geometric structures in statistical physics · 2017. 7. 17. · they are fundamental in the...
TRANSCRIPT
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Geometric structures in statistical physics:
surfaces, amoebas and tropical limit
Mario Angelelli
Department of Mathematics and Physics “E. De Giorgi”, University of Salento & INFN, 73100 Lecce (IT)
Ph.D. course in Physics and Nanosciences - XXIX cycle
Advisor: Prof. B. G. KonopelchenkoCo-Advisor: Prof. G. Landolfi
Coordinator of the Ph. D. Course: Prof. R. Rinaldi
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Introduction: statement of the problem
Partition function and free energy are fundamental concepts in physics:
They encode the physical information of complex systems, balancing statistical datawith physical requirements (e.g. observed values of macroscopic quantities).
They allow to derive physical characteristics (e.g., generating correlation functions).
They are fundamental in the description of several ensembles in statistical physics.
They have a large number of extensions: discrete (usual sum), countable (series),continuous (integral), generalizations to quantum settings (trace operator algebras)and field theories (functional integration).
Now they are applied in many branches of science, e.g. mathematics, biology,information theory, economics, AI, . . .
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Introduction: statement of the problem
Partition function and free energy are fundamental concepts in physics:
They encode the physical information of complex systems, balancing statistical datawith physical requirements (e.g. observed values of macroscopic quantities).
They allow to derive physical characteristics (e.g., generating correlation functions).
They are fundamental in the description of several ensembles in statistical physics.
They have a large number of extensions: discrete (usual sum), countable (series),continuous (integral), generalizations to quantum settings (trace operator algebras)and field theories (functional integration).
Now they are applied in many branches of science, e.g. mathematics, biology,information theory, economics, AI, . . .
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Introduction: statement of the problem
Partition function and free energy are fundamental concepts in physics:
They encode the physical information of complex systems, balancing statistical datawith physical requirements (e.g. observed values of macroscopic quantities).
They allow to derive physical characteristics (e.g., generating correlation functions).
They are fundamental in the description of several ensembles in statistical physics.
They have a large number of extensions: discrete (usual sum), countable (series),continuous (integral), generalizations to quantum settings (trace operator algebras)and field theories (functional integration).
Now they are applied in many branches of science, e.g. mathematics, biology,information theory, economics, AI, . . .
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Introduction: statement of the problem
!!! Cases of explicit evaluation of partition function are very rare, due to severalcomplications: (exponentially) hard computations, occurrence of collective phenomena,dependence from boundary conditions, sensitivity to initial conditions for dynamicalsystems, random behaviour etc.
I Our approach: extract distinctive features from partition function and free energy,figure out main characteristics using geometry, investigate composition using algebra.
Goals of the Ph.D. project
Extract relevant features of P.F. and F.E. for a large class of systems, in order notto rely on particular features of specific models;
provide simple tools to explore and characterize relevant physical properties (e.g.,role of interactions, symmetries);
identify phenomena more suitable for this description;
connect di↵erent techniques and contexts that share similarities with thisdescription; applications beyond statistical physics (e.g., integrable systems).
For more details on activities during the Ph. D. course, see ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Introduction: statement of the problem
!!! Cases of explicit evaluation of partition function are very rare, due to severalcomplications: (exponentially) hard computations, occurrence of collective phenomena,dependence from boundary conditions, sensitivity to initial conditions for dynamicalsystems, random behaviour etc.
I Our approach: extract distinctive features from partition function and free energy,figure out main characteristics using geometry, investigate composition using algebra.
Goals of the Ph.D. project
Extract relevant features of P.F. and F.E. for a large class of systems, in order notto rely on particular features of specific models;
provide simple tools to explore and characterize relevant physical properties (e.g.,role of interactions, symmetries);
identify phenomena more suitable for this description;
connect di↵erent techniques and contexts that share similarities with thisdescription; applications beyond statistical physics (e.g., integrable systems).
For more details on activities during the Ph. D. course, see ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Introduction: statement of the problem
!!! Cases of explicit evaluation of partition function are very rare, due to severalcomplications: (exponentially) hard computations, occurrence of collective phenomena,dependence from boundary conditions, sensitivity to initial conditions for dynamicalsystems, random behaviour etc.
I Our approach: extract distinctive features from partition function and free energy,figure out main characteristics using geometry, investigate composition using algebra.
Goals of the Ph.D. project
Extract relevant features of P.F. and F.E. for a large class of systems, in order notto rely on particular features of specific models;
provide simple tools to explore and characterize relevant physical properties (e.g.,role of interactions, symmetries);
identify phenomena more suitable for this description;
connect di↵erent techniques and contexts that share similarities with thisdescription; applications beyond statistical physics (e.g., integrable systems).
For more details on activities during the Ph. D. course, see ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Geometry and statistical mappings
Geometric descriptions of statistical objects: statistical hypersurfaces and amoebas.
Tropical limit in statistical physics
Tropical algebra: definitions, applications to statistical physics. How to extractunderlying tropical structures in statistical models and formalize them.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Geometry and statistical mappings
Geometric descriptions of statistical objects: statistical hypersurfaces and amoebas.
Tropical limit in statistical physics
Tropical algebra: definitions, applications to statistical physics. How to extractunderlying tropical structures in statistical models and formalize them.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
How to identify and, then,relate the microscopic andmacroscopic sectors?
How to describe more generalstatistical “models”beyond thecanonical ensemble?
Once interactions areincluded, how can they bestudied?
Desiderata
So we need to take more spaces and a mappingbetween them where the microscopic sector dependson a set of parameters and with extra-structures,e.g. metric, which measure deviation from an idealmodel.
Perspective
Focus on relations between spaces and see how theirinformation contents are related.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
How to identify and, then,relate the microscopic andmacroscopic sectors?
How to describe more generalstatistical “models”beyond thecanonical ensemble?
Once interactions areincluded, how can they bestudied?
Desiderata
So we need to take more spaces and a mappingbetween them
where the microscopic sector dependson a set of parameters and with extra-structures,e.g. metric, which measure deviation from an idealmodel.
Perspective
Focus on relations between spaces and see how theirinformation contents are related.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
How to identify and, then,relate the microscopic andmacroscopic sectors?
How to describe more generalstatistical “models”beyond thecanonical ensemble?
Once interactions areincluded, how can they bestudied?
Desiderata
So we need to take more spaces and a mappingbetween them
where the microscopic sector dependson a set of parameters and with extra-structures,e.g. metric, which measure deviation from an idealmodel.
Perspective
Focus on relations between spaces and see how theirinformation contents are related.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
How to identify and, then,relate the microscopic andmacroscopic sectors?
How to describe more generalstatistical “models”beyond thecanonical ensemble?
Once interactions areincluded, how can they bestudied?
Desiderata
So we need to take more spaces and a mappingbetween them where the microscopic sector dependson a set of parameters
and with extra-structures,e.g. metric, which measure deviation from an idealmodel.
Perspective
Focus on relations between spaces and see how theirinformation contents are related.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
How to identify and, then,relate the microscopic andmacroscopic sectors?
How to describe more generalstatistical “models”beyond thecanonical ensemble?
Once interactions areincluded, how can they bestudied?
Desiderata
So we need to take more spaces and a mappingbetween them where the microscopic sector dependson a set of parameters
and with extra-structures,e.g. metric, which measure deviation from an idealmodel.
Perspective
Focus on relations between spaces and see how theirinformation contents are related.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
How to identify and, then,relate the microscopic andmacroscopic sectors?
How to describe more generalstatistical “models”beyond thecanonical ensemble?
Once interactions areincluded, how can they bestudied?
Desiderata
So we need to take more spaces and a mappingbetween them where the microscopic sector dependson a set of parameters and with extra-structures,e.g. metric, which measure deviation from an idealmodel.
Perspective
Focus on relations between spaces and see how theirinformation contents are related.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
How to identify and, then,relate the microscopic andmacroscopic sectors?
How to describe more generalstatistical “models”beyond thecanonical ensemble?
Once interactions areincluded, how can they bestudied?
Desiderata
So we need to take more spaces and a mappingbetween them where the microscopic sector dependson a set of parameters and with extra-structures,e.g. metric, which measure deviation from an idealmodel.
Perspective
Focus on relations between spaces and see how theirinformation contents are related.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Free-energy-oriented approach: basic statistical mappings
We realize the relation between spaces as a mapping relating their information contents.
Physical insight from partition function Z and free energy F
F = �kB
T · lnZ = �kB
T ln
0
@X
{n}
e�
E{n}k
B
T
1
A . (1)
I Extend free energy properties to a more general class of statistical mappingsintroducting f↵(x)
F (x1, . . . , xn) = ln
NX
↵=1
e f↵(x1,...,xn)!. (2)
Geometry
(2) can be seen as a n-dimensional hypersurface (statistical hypersurface) embedded ina (n + 1)-dimensional ambient space. If the ambient space is endowed with a metricstructure, this hypersurface inherits metric properties.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Free-energy-oriented approach: basic statistical mappings
We realize the relation between spaces as a mapping relating their information contents.Physical insight from partition function Z and free energy F
F = �kB
T · lnZ = �kB
T ln
0
@X
{n}
e�
E{n}k
B
T
1
A . (1)
I Extend free energy properties to a more general class of statistical mappingsintroducting f↵(x)
F (x1, . . . , xn) = ln
NX
↵=1
e f↵(x1,...,xn)!. (2)
Geometry
(2) can be seen as a n-dimensional hypersurface (statistical hypersurface) embedded ina (n + 1)-dimensional ambient space. If the ambient space is endowed with a metricstructure, this hypersurface inherits metric properties.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Free-energy-oriented approach: basic statistical mappings
We realize the relation between spaces as a mapping relating their information contents.Physical insight from partition function Z and free energy F
F = �kB
T · lnZ = �kB
T ln
0
@X
{n}
e�
E{n}k
B
T
1
A . (1)
I Extend free energy properties to a more general class of statistical mappingsintroducting f↵(x)
F (x1, . . . , xn) = ln
NX
↵=1
e f↵(x1,...,xn)!. (2)
Geometry
(2) can be seen as a n-dimensional hypersurface (statistical hypersurface) embedded ina (n + 1)-dimensional ambient space. If the ambient space is endowed with a metricstructure, this hypersurface inherits metric properties.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Free-energy-oriented approach: basic statistical mappings
We realize the relation between spaces as a mapping relating their information contents.Physical insight from partition function Z and free energy F
F = �kB
T · lnZ = �kB
T ln
0
@X
{n}
e�
E{n}k
B
T
1
A . (1)
I Extend free energy properties to a more general class of statistical mappingsintroducting f↵(x)
F (x1, . . . , xn) = ln
NX
↵=1
e f↵(x1,...,xn)!. (2)
Geometry
(2) can be seen as a n-dimensional hypersurface (statistical hypersurface) embedded ina (n + 1)-dimensional ambient space. If the ambient space is endowed with a metricstructure, this hypersurface inherits metric properties.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Main advantages
Free energy-focused approach: it gives a clear picture of its geometriccharacteristics for a wide class of models.
Alternative approach w.r.t. information geometry (or other statistical manifolds),that is oriented towards statistical inference ?
. . . so it is more suited to relevant parametrizations of physical systems (e.g.super-ideal mappings are non-degenerate).
It identifies characterizing objects: Gauss-Kronecker curvature.
Figure: Plane and cylinder are both flat, but the ways they are embedded in R3 are di↵erent: they have di↵erent shapes. TheGauss-Kronecker (G-K) curvature (determinant of the shape operator) is an intrinsic description of immersed hypersurfaces (upto (�1)n).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Main advantages
Free energy-focused approach: it gives a clear picture of its geometriccharacteristics for a wide class of models.
Alternative approach w.r.t. information geometry (or other statistical manifolds),that is oriented towards statistical inference ?
. . . so it is more suited to relevant parametrizations of physical systems (e.g.super-ideal mappings are non-degenerate).
It identifies characterizing objects: Gauss-Kronecker curvature.
Figure: Plane and cylinder are both flat, but the ways they are embedded in R3 are di↵erent: they have di↵erent shapes. TheGauss-Kronecker (G-K) curvature (determinant of the shape operator) is an intrinsic description of immersed hypersurfaces (upto (�1)n).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Main advantages
Free energy-focused approach: it gives a clear picture of its geometriccharacteristics for a wide class of models.
Alternative approach w.r.t. information geometry (or other statistical manifolds),that is oriented towards statistical inference ?
. . . so it is more suited to relevant parametrizations of physical systems (e.g.super-ideal mappings are non-degenerate).
It identifies characterizing objects: Gauss-Kronecker curvature.
Figure: Plane and cylinder are both flat, but the ways they are embedded in R3 are di↵erent: they have di↵erent shapes. TheGauss-Kronecker (G-K) curvature (determinant of the shape operator) is an intrinsic description of immersed hypersurfaces (upto (�1)n).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Main advantages
Free energy-focused approach: it gives a clear picture of its geometriccharacteristics for a wide class of models.
Alternative approach w.r.t. information geometry (or other statistical manifolds),that is oriented towards statistical inference ?
. . . so it is more suited to relevant parametrizations of physical systems (e.g.super-ideal mappings are non-degenerate).
It identifies characterizing objects: Gauss-Kronecker curvature.
Figure: Plane and cylinder are both flat, but the ways they are embedded in R3 are di↵erent: they have di↵erent shapes. TheGauss-Kronecker (G-K) curvature (determinant of the shape operator) is an intrinsic description of immersed hypersurfaces (upto (�1)n).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Basic statistical mappings: main results
For statistical hypersurfaces, geometric characteristics are linked to probability andassociated moments: they depend on averages Q :=
P↵ w↵Q↵ and correlations
Q,P :=P
↵ w↵Q↵P↵ w.r.t. Gibbs distribution w↵ := ef↵(x)�F (x).
Main results: linear f↵(x) = b↵ +nX
i=1
a↵ixi
KG�K = 0 i↵ there exists ~x0 2 Rn : (a · ~x0)` is independent of ` = 1, . . . ,N. If the
G-K curvature vanishes, then there exists a Killing vector field K :=nX
i=1
ci
@@x
i
,
c1, . . . , cn constants.
For a super-ideal statistical hypersurface xn+1 = ln
nX
i=1
exi
!G-K curvature
vanishes, while mean and scalar curvatures take values in
0 ⌦ n � 1pn(n + 1)
, 0 R (n � 1)(n � 2)n(n + 1)
. (3)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Basic statistical mappings: main results
For statistical hypersurfaces, geometric characteristics are linked to probability andassociated moments: they depend on averages Q :=
P↵ w↵Q↵ and correlations
Q,P :=P
↵ w↵Q↵P↵ w.r.t. Gibbs distribution w↵ := ef↵(x)�F (x).
Main results: linear f↵(x) = b↵ +nX
i=1
a↵ixi
KG�K = 0 i↵ there exists ~x0 2 Rn : (a · ~x0)` is independent of ` = 1, . . . ,N. If the
G-K curvature vanishes, then there exists a Killing vector field K :=nX
i=1
ci
@@x
i
,
c1, . . . , cn constants.
For a super-ideal statistical hypersurface xn+1 = ln
nX
i=1
exi
!G-K curvature
vanishes, while mean and scalar curvatures take values in
0 ⌦ n � 1pn(n + 1)
, 0 R (n � 1)(n � 2)n(n + 1)
. (3)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Basic statistical mappings: main results
For statistical hypersurfaces, geometric characteristics are linked to probability andassociated moments: they depend on averages Q :=
P↵ w↵Q↵ and correlations
Q,P :=P
↵ w↵Q↵P↵ w.r.t. Gibbs distribution w↵ := ef↵(x)�F (x).
Main results: linear f↵(x) = b↵ +nX
i=1
a↵ixi
KG�K = 0 i↵ there exists ~x0 2 Rn : (a · ~x0)` is independent of ` = 1, . . . ,N. If the
G-K curvature vanishes, then there exists a Killing vector field K :=nX
i=1
ci
@@x
i
,
c1, . . . , cn constants.
For a super-ideal statistical hypersurface xn+1 = ln
nX
i=1
exi
!G-K curvature
vanishes, while mean and scalar curvatures take values in
0 ⌦ n � 1pn(n + 1)
, 0 R (n � 1)(n � 2)n(n + 1)
. (3)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Main results on statistical hypersurfaces
Main physical ideal models have vanishing of G-K curvature. The converse is not true:also non-linear f↵ can give vanishing G-K. It is related to the invariance w.r.t. choice ofthe“ground energy”. If it holds also for independent variables x 7! x+ C thenK
G�K = 0.
Main results: non-linear f↵
General formulas are provided for metric objects ? .
Entropy is expressed as a coupling between the hypersurface and its normal bundle.For homogeneous interactions of degree d
S =p
det g�!X ·�!N + (d � 1)f̄ (4)
Metric on the hyper-surface
Position vector(x1, . . . , xn,F )
Unit normal vectorwith a given orienta-tion
mean value f̄ =PN
↵=1 w↵f↵
Metric characterization of s.-i. mapping: under the rank and the closure hypotheses
g̃ij
(x, t) = �ij
+ hi
(x, t) · hj
(x, t),@h
i
@xj
= C · (�ij
hj
� hi
hj
) (5)
(t deformation parameter) there exists a potential function F̃ such that hi
= @x
i
F̃and F̃ (x, t) = log(�(t) +
Pn
i=1 ex
i
�↵i
(t)).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Main results on statistical hypersurfaces
Main physical ideal models have vanishing of G-K curvature. The converse is not true:also non-linear f↵ can give vanishing G-K. It is related to the invariance w.r.t. choice ofthe“ground energy”. If it holds also for independent variables x 7! x+ C thenK
G�K = 0.Main results: non-linear f↵
General formulas are provided for metric objects ? .
Entropy is expressed as a coupling between the hypersurface and its normal bundle.For homogeneous interactions of degree d
S =p
det g�!X ·�!N + (d � 1)f̄ (4)
Metric on the hyper-surface
Position vector(x1, . . . , xn,F )
Unit normal vectorwith a given orienta-tion
mean value f̄ =PN
↵=1 w↵f↵
Metric characterization of s.-i. mapping: under the rank and the closure hypotheses
g̃ij
(x, t) = �ij
+ hi
(x, t) · hj
(x, t),@h
i
@xj
= C · (�ij
hj
� hi
hj
) (5)
(t deformation parameter) there exists a potential function F̃ such that hi
= @x
i
F̃and F̃ (x, t) = log(�(t) +
Pn
i=1 ex
i
�↵i
(t)).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Main results on statistical hypersurfaces
Main physical ideal models have vanishing of G-K curvature. The converse is not true:also non-linear f↵ can give vanishing G-K. It is related to the invariance w.r.t. choice ofthe“ground energy”. If it holds also for independent variables x 7! x+ C thenK
G�K = 0.Main results: non-linear f↵
General formulas are provided for metric objects ? .
Entropy is expressed as a coupling between the hypersurface and its normal bundle.For homogeneous interactions of degree d
S =p
det g�!X ·�!N + (d � 1)f̄ (4)
Metric on the hyper-surface
Position vector(x1, . . . , xn,F )
Unit normal vectorwith a given orienta-tion
mean value f̄ =PN
↵=1 w↵f↵
Metric characterization of s.-i. mapping: under the rank and the closure hypotheses
g̃ij
(x, t) = �ij
+ hi
(x, t) · hj
(x, t),@h
i
@xj
= C · (�ij
hj
� hi
hj
) (5)
(t deformation parameter) there exists a potential function F̃ such that hi
= @x
i
F̃and F̃ (x, t) = log(�(t) +
Pn
i=1 ex
i
�↵i
(t)).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Main results on statistical hypersurfaces
Main physical ideal models have vanishing of G-K curvature. The converse is not true:also non-linear f↵ can give vanishing G-K. It is related to the invariance w.r.t. choice ofthe“ground energy”. If it holds also for independent variables x 7! x+ C thenK
G�K = 0.Main results: non-linear f↵
General formulas are provided for metric objects ? .
Entropy is expressed as a coupling between the hypersurface and its normal bundle.For homogeneous interactions of degree d
S =p
det g�!X ·�!N + (d � 1)f̄ (4)
Metric on the hyper-surface
Position vector(x1, . . . , xn,F )
Unit normal vectorwith a given orienta-tion
mean value f̄ =PN
↵=1 w↵f↵
Metric characterization of s.-i. mapping: under the rank and the closure hypotheses
g̃ij
(x, t) = �ij
+ hi
(x, t) · hj
(x, t),@h
i
@xj
= C · (�ij
hj
� hi
hj
) (5)
(t deformation parameter) there exists a potential function F̃ such that hi
= @x
i
F̃and F̃ (x, t) = log(�(t) +
Pn
i=1 ex
i
�↵i
(t)).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Main results on statistical hypersurfaces
Main physical ideal models have vanishing of G-K curvature. The converse is not true:also non-linear f↵ can give vanishing G-K. It is related to the invariance w.r.t. choice ofthe“ground energy”. If it holds also for independent variables x 7! x+ C thenK
G�K = 0.Main results: non-linear f↵
General formulas are provided for metric objects ? .
Entropy is expressed as a coupling between the hypersurface and its normal bundle.For homogeneous interactions of degree d
S =p
det g�!X ·�!N + (d � 1)f̄ (4)
Metric on the hyper-surface
Position vector(x1, . . . , xn,F )
Unit normal vectorwith a given orienta-tion
mean value f̄ =PN
↵=1 w↵f↵
Metric characterization of s.-i. mapping: under the rank and the closure hypotheses
g̃ij
(x, t) = �ij
+ hi
(x, t) · hj
(x, t),@h
i
@xj
= C · (�ij
hj
� hi
hj
) (5)
(t deformation parameter) there exists a potential function F̃ such that hi
= @x
i
F̃and F̃ (x, t) = log(�(t) +
Pn
i=1 ex
i
�↵i
(t)).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some results on statistical hypersurfaces
In case of perturbations of P ideal subsystems due to "⌧ 1, Gauss-Kroneckercurvature still vanishes at first order in ". Higher orders are in generalnon-vanishing.
Several examples of non-linear interactions are provided, both with vanishing G-Kcurvature (cilindrical-type symmetry) and not.
Phase singularities are described by singularities of the metric. They can behidden or visible from the curvature p.o.v., depending on the specific form of f↵s.There can be first-order phase transitions (singularity of the metric) and higherorder (singularity of the curvature). Characterizing cases are explored.
For more details, see also [A. & Konopelchenko, 2016]
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some results on statistical hypersurfaces
In case of perturbations of P ideal subsystems due to "⌧ 1, Gauss-Kroneckercurvature still vanishes at first order in ". Higher orders are in generalnon-vanishing.
Several examples of non-linear interactions are provided, both with vanishing G-Kcurvature (cilindrical-type symmetry) and not.
Phase singularities are described by singularities of the metric. They can behidden or visible from the curvature p.o.v., depending on the specific form of f↵s.There can be first-order phase transitions (singularity of the metric) and higherorder (singularity of the curvature). Characterizing cases are explored.
For more details, see also [A. & Konopelchenko, 2016]
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some results on statistical hypersurfaces
In case of perturbations of P ideal subsystems due to "⌧ 1, Gauss-Kroneckercurvature still vanishes at first order in ". Higher orders are in generalnon-vanishing.
Several examples of non-linear interactions are provided, both with vanishing G-Kcurvature (cilindrical-type symmetry) and not.
Phase singularities are described by singularities of the metric. They can behidden or visible from the curvature p.o.v., depending on the specific form of f↵s.There can be first-order phase transitions (singularity of the metric) and higherorder (singularity of the curvature). Characterizing cases are explored.
For more details, see also [A. & Konopelchenko, 2016]
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Statistical amoebas: motivations
More subsystems:
N1X
↵=1
eg↵(x) = e fN1+N2+1(x) =N2X
�=1
eh� (x). (6)
Relaxed (physical) assumption on g↵: negative degeneracies of energy levels,negative probabilities [Wigner, 1932; Dirac, 1942; Feynman, 1987; Blizard,1990; Burgin, 2010]).
Singularity of free energy at the zeros of the partition function , phase transition[Lee & Yang, 1952; Lee & Yang, 1952, b; Newman & Schulman, 1980].
+Real non-positive definite partition functions combine reality constraints (typical ofthermodynamics) with the investigation of metastable states in terms of zeros of P.F.
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Statistical amoebas: motivations
More subsystems:
N1X
↵=1
eg↵(x) =
e fN1+N2+1(x) =
N2X
�=1
eh� (x). (6)
Relaxed (physical) assumption on g↵: negative degeneracies of energy levels,negative probabilities [Wigner, 1932; Dirac, 1942; Feynman, 1987; Blizard,1990; Burgin, 2010]).
Singularity of free energy at the zeros of the partition function , phase transition[Lee & Yang, 1952; Lee & Yang, 1952, b; Newman & Schulman, 1980].
+Real non-positive definite partition functions combine reality constraints (typical ofthermodynamics) with the investigation of metastable states in terms of zeros of P.F.
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IntroductionGeometric micro-macro mappings
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Statistical amoebas: motivations
More subsystems:
0 = �N1X
↵=1
e f↵(x) +N2X
�=1
e f� (x). (6)
Relaxed (physical) assumption on g↵: negative degeneracies of energy levels,negative probabilities [Wigner, 1932; Dirac, 1942; Feynman, 1987; Blizard,1990; Burgin, 2010]).
Singularity of free energy at the zeros of the partition function , phase transition[Lee & Yang, 1952; Lee & Yang, 1952, b; Newman & Schulman, 1980].
+Real non-positive definite partition functions combine reality constraints (typical ofthermodynamics) with the investigation of metastable states in terms of zeros of P.F.
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: motivations
More subsystems:
0 = �N1X
↵=1
e f↵(x) +N2X
�=1
e f� (x). (6)
Relaxed (physical) assumption on g↵: negative degeneracies of energy levels,negative probabilities [Wigner, 1932; Dirac, 1942; Feynman, 1987; Blizard,1990; Burgin, 2010]).
Singularity of free energy at the zeros of the partition function , phase transition[Lee & Yang, 1952; Lee & Yang, 1952, b; Newman & Schulman, 1980].
+Real non-positive definite partition functions combine reality constraints (typical ofthermodynamics) with the investigation of metastable states in terms of zeros of P.F.
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: motivations
More subsystems:
0 = �N1X
↵=1
e f↵(x) +N2X
�=1
e f� (x). (6)
Relaxed (physical) assumption on g↵: negative degeneracies of energy levels,negative probabilities [Wigner, 1932; Dirac, 1942; Feynman, 1987; Blizard,1990; Burgin, 2010]).
Singularity of free energy at the zeros of the partition function , phase transition[Lee & Yang, 1952; Lee & Yang, 1952, b; Newman & Schulman, 1980].
+Real non-positive definite partition functions combine reality constraints (typical ofthermodynamics) with the investigation of metastable states in terms of zeros of P.F.
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: motivations
More subsystems:
0 = �N1X
↵=1
e f↵(x) +N2X
�=1
e f� (x). (6)
Relaxed (physical) assumption on g↵: negative degeneracies of energy levels,negative probabilities [Wigner, 1932; Dirac, 1942; Feynman, 1987; Blizard,1990; Burgin, 2010]).
Singularity of free energy at the zeros of the partition function , phase transition[Lee & Yang, 1952; Lee & Yang, 1952, b; Newman & Schulman, 1980].
+Real non-positive definite partition functions combine reality constraints (typical ofthermodynamics) with the investigation of metastable states in terms of zeros of P.F.
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
I 7! {1, . . . ,N}\I acts asg↵ 7! �g↵ and Zk(I) 7! �Zk(I).One can fix #I N �#I toavoid such a redundancy.
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
I 7! {1, . . . ,N}\I acts asg↵ 7! �g↵ and Zk(I) 7! �Zk(I).One can fix #I N �#I toavoid such a redundancy.
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
Hypersurfaces (7) divide Rn indomains. In open subdomains, allfunctions Z
k
(I) have well-defined(positive or negative) sign.
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
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Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
Hypersurfaces (7) divide Rn indomains. In open subdomains, allfunctions Z
k
(I) have well-defined(positive or negative) sign.
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
One can fix an order (e.g.,lexicographic) on k-subsets of [N]and associate with each subdomaina vector
(Sk;�)⌧ := sign
Z(I⌧ ; x) : 1 ⌧
Nk
!!
(8)evaluated at an interior point x.
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Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
One can fix an order (e.g.,lexicographic) on k-subsets of [N]and associate with each subdomaina vector
(Sk;�)⌧ := sign
Z(I⌧ ; x) : 1 ⌧
Nk
!!
(8)evaluated at an interior point x.
Mario Angelelli Geometric structures in statistical physics
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Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
Stability domain Dk+ ✓ Rn:
Zk
(I) > 0 for all I with #I = k.Instability domain D
k�: maximalnumber of negative Z
k
(I).Hypersurfaces of the k-stratum areconfined in the zeros confinementdomain
ZCDk
:= Rn\(Dk+ [Dk�).
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
Stability domain Dk+ ✓ Rn:
Zk
(I) > 0 for all I with #I = k.Instability domain D
k�: maximalnumber of negative Z
k
(I).Hypersurfaces of the k-stratum areconfined in the zeros confinementdomain
ZCDk
:= Rn\(Dk+ [Dk�).
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
The set Dk+ [ ZCDk is the
statistical k-amoeba and itsboundary contains its extremalpoints.
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Statistical amoebas: definitions
If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce
Z(I; x) := �X
↵2I
e f↵(x) +X
�/2I
e f� (x) = 0. (7)
The set Dk+ [ ZCDk is the
statistical k-amoeba and itsboundary contains its extremalpoints.
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Statistical amoebas: main results for polymonial f↵
Theorem
The maximum number of �1 in Sk
at fixed k <N2
is equal to�N�1k�1�. If 2k = N then
the number of �1 signs in SN
2is identically equal to
�2k�1
k
�on Rn\Z
sing, N2. Then, the
integral characteristic
S̄
k;� =1�N
k
� ·(Nk
)X
⌧=1
(Sk;�)⌧ (8)
takes values in the interval⇥1� 2 k
N
; 1⇤. The maximum 1 (resp. minimum �1 + 2 k
N
) isreached in the stability (resp. instability) domain D
k+ (resp. Dk�).
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Statistical amoebas: main results for polymonial f↵
Theorem
The maximum number of �1 in Sk
at fixed k <N2
is equal to�N�1k�1�. If 2k = N then
the number of �1 signs in SN
2is identically equal to
�2k�1
k
�on Rn\Z
sing, N2. Then, the
integral characteristic
S̄
k;� =1�N
k
� ·(Nk
)X
⌧=1
(Sk;�)⌧ (8)
takes values in the interval⇥1� 2 k
N
; 1⇤. The maximum 1 (resp. minimum �1 + 2 k
N
) isreached in the stability (resp. instability) domain D
k+ (resp. Dk�).
Proposition
If 1 k < k̂ �N2
⌫, then
Dk̂+ ✓ Dk+, Dk� ✓ Dk̂�. (9)
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Statistical amoebas: main results for polymonial f↵
Proposition
If 1 k < k̂ �N2
⌫, then
Dk̂+ ✓ Dk+, Dk� ✓ Dk̂�. (8)
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Statistical amoebas: main results
Lemma
Two hypersurfaces (7) belonging to the same stratum Zsing,k and di↵erent I1 and I2
intersect at finite x only if I1 \ I2 6= ;.
Proposition
If {f↵(x) : ↵ 2 [N]} are pairwise di↵erent polynomial functions, then Zsing,k̂ lies inside a
region of Rn delimited by some components of Zsing,k , 1 k < k̂
�N2
⌫.
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Statistical amoebas: main results
Proposition
The equilibrium domain Dk+ is the set of all points x such that, for all
N � k + 1 g N, there exists a non-degenerate polygon with g sides of lengthsX↵2I1
e f↵(x), . . . ,X
↵2Ig
e f↵(x) where {I1, . . . , Ig} is any partition of [N] in non-empty
subsets.
Proposition
For homogeneous functions, connected components of Dk� are unbounded for all
1 k N2
� 1.
Spin interpretation: “microcanonical”probability wN,k =
�N
k
��1on each domain,
“canonical”probability with energy (8).
Connections with algebraic geometry: s.a. generalize (non-)lopsided sets, whichare good approximations of an algebraic amoeba. The introduction of amoebastakes us to the realm of tropical mathematics.
For more details, see also [A. & Konopelchenko, 2016, b]
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Statistical amoebas: main results
Proposition
The equilibrium domain Dk+ is the set of all points x such that, for all
N � k + 1 g N, there exists a non-degenerate polygon with g sides of lengthsX↵2I1
e f↵(x), . . . ,X
↵2Ig
e f↵(x) where {I1, . . . , Ig} is any partition of [N] in non-empty
subsets.
Proposition
For homogeneous functions, connected components of Dk� are unbounded for all
1 k N2
� 1.
Spin interpretation: “microcanonical”probability wN,k =
�N
k
��1on each domain,
“canonical”probability with energy (8).
Connections with algebraic geometry: s.a. generalize (non-)lopsided sets, whichare good approximations of an algebraic amoeba. The introduction of amoebastakes us to the realm of tropical mathematics.
For more details, see also [A. & Konopelchenko, 2016, b]
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Geometry and statistical mappings
Geometric descriptions of statistical objects: statistical hypersurfaces and amoebas.
Tropical limit in statistical physics
Tropical algebra: definitions, applications to statistical physics. How to extractunderlying tropical structures in statistical models and formalize them.
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Tropical limit in statistical physics
Tropical limit: introduction
Tropical limit: transformation which results from a limit procedure with an essentialsingularity ? .
Physical rationale: study of objects for small values of characterizing certain“constants”(! deformations). Ex: semiclassical limit of a wavefunction
(x, t) = A(x, t) · exp✓iS(x, t)
~
◆
at ~ ! 0.
(9)
Semiclassical limit of awavefunction
Linear di↵erential equations generally becomes non-linear in this limit (e.g.Schrödinger Hamilton-Jacobi) and the superposition principle is lost unless algebraicoperations change too!
z =NX
`=1
x` eZ"/" =
NX
`=1
eX`,"/". (10)Family of “additions”
At "! 0+: semiring ?(11)
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limit: introduction
Tropical limit: transformation which results from a limit procedure with an essentialsingularity ? .Physical rationale: study of objects for small values of characterizing certain“constants”(! deformations). Ex: semiclassical limit of a wavefunction
(x, t) = A(x, t) · exp✓iS(x, t)
~
◆at ~ ! 0. (9)
Semiclassical limit of awavefunction
Linear di↵erential equations generally becomes non-linear in this limit (e.g.Schrödinger Hamilton-Jacobi) and the superposition principle is lost unless algebraicoperations change too!
z =NX
`=1
x` eZ"/" =
NX
`=1
eX`,"/". (10)Family of “additions”
At "! 0+: semiring ?(11)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limit: introduction
Tropical limit: transformation which results from a limit procedure with an essentialsingularity ? .Physical rationale: study of objects for small values of characterizing certain“constants”(! deformations). Ex: semiclassical limit of a wavefunction
(x, t) =
A(x, t) ·
exp
✓
i
S(x, t)~
◆at ~ ! 0. (9)
Semiclassical limit of awavefunction
Linear di↵erential equations generally becomes non-linear in this limit (e.g.Schrödinger Hamilton-Jacobi) and the superposition principle is lost unless algebraicoperations change too!
z =NX
`=1
x` eZ"/" =
NX
`=1
eX`,"/". (10)Family of “additions”
At "! 0+: semiring ?(11)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limit: introduction
Tropical limit: transformation which results from a limit procedure with an essentialsingularity ? .Physical rationale: study of objects for small values of characterizing certain“constants”(! deformations). Ex: semiclassical limit of a wavefunction
(x, t) =
A(x, t) ·
exp
✓
i
S(x, t)~
◆at ~ ! 0. (9)
Semiclassical limit of awavefunction
Linear di↵erential equations generally becomes non-linear in this limit (e.g.Schrödinger Hamilton-Jacobi) and the superposition principle is lost unless algebraicoperations change too!
z =NX
`=1
x` eZ"/" =
NX
`=1
eX`,"/". (10)Family of “additions”
At "! 0+: semiring ?(11)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limit: introduction
Tropical limit: transformation which results from a limit procedure with an essentialsingularity ? .Physical rationale: study of objects for small values of characterizing certain“constants”(! deformations). Ex: semiclassical limit of a wavefunction
(x, t) =
A(x, t) ·
exp
✓
i
S(x, t)~
◆at ~ ! 0. (9)
Semiclassical limit of awavefunction
Linear di↵erential equations generally becomes non-linear in this limit (e.g.Schrödinger Hamilton-Jacobi) and the superposition principle is lost unless algebraicoperations change too!
z =NX
`=1
x` eZ"/" =
NX
`=1
eX`,"/". (10)Family of “additions”
At "! 0+: semiring ?
Xtrop
� Ytrop
:= lim"!0
✓" log
✓exp
Xtrop
"· exp Ytrop
"
◆◆= X
trop
+ Ytrop
, (11)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limit: introduction
Tropical limit: transformation which results from a limit procedure with an essentialsingularity ? .Physical rationale: study of objects for small values of characterizing certain“constants”(! deformations). Ex: semiclassical limit of a wavefunction
(x, t) =
A(x, t) ·
exp
✓
i
S(x, t)~
◆at ~ ! 0. (9)
Semiclassical limit of awavefunction
Linear di↵erential equations generally becomes non-linear in this limit (e.g.Schrödinger Hamilton-Jacobi) and the superposition principle is lost unless algebraicoperations change too!
z =NX
`=1
x` eZ"/" =
NX
`=1
eX`,"/". (10)Family of “additions”
At "! 0+: semiring ?
Xtrop
� Ytrop
:= lim"!0
✓" log
✓exp
Xtrop
"+ exp
Ytrop
"
◆◆= max{X
trop
,Ytrop
}. (11)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limit: introduction
Tropical limit: transformation which results from a limit procedure with an essentialsingularity ? .Physical rationale: study of objects for small values of characterizing certain“constants”(! deformations). Ex: semiclassical limit of a wavefunction
(x, t) =
A(x, t) ·
exp
✓
i
S(x, t)~
◆at ~ ! 0. (9)
Semiclassical limit of awavefunction
Linear di↵erential equations generally becomes non-linear in this limit (e.g.Schrödinger Hamilton-Jacobi) and the superposition principle is lost unless algebraicoperations change too!
z =NX
`=1
x` eZ"/" =
NX
`=1
eX`,"/". (10)Family of “additions”
At "! 0+: semiring with idempotent addition: a� a = max{a, a} = a ?
Xtrop
� Ytrop
:= lim"!0
✓" log
✓exp
Xtrop
"+ exp
Ytrop
"
◆◆= max{X
trop
,Ytrop
}. (11)
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Tropical limit in statistical physics
Tropical limit in statistical physics
Our starting point is
kB
! 0. (12)
In more details,
Definition
Given a physical statistical system specified by a partition function and an associatedfree energy, its tropical limit is defined as the simultaneous limit for k
B
and Avogadronumber N
A
such that their product is kept constant, i.e.
kB
! 0+, NA
! 1 : kB
· NA
=: R is constant.
F = �kB
T lnX
n
gn
exp
✓� EnkB
T
◆�! F
trop
(T ) = �TM
n
✓�Fn
T
◆
g
n
=: exp
S
n
k
B
!,
F
n
:= En
� T · Sn
%
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limit in statistical physics
Our starting point is
kB
! 0. (12)
In more details,
Definition
Given a physical statistical system specified by a partition function and an associatedfree energy, its tropical limit is defined as the simultaneous limit for k
B
and Avogadronumber N
A
such that their product is kept constant, i.e.
kB
! 0+, NA
! 1 : kB
· NA
=: R is constant.
F = �kB
T lnX
n
gn
exp
✓� EnkB
T
◆�! F
trop
(T ) = �TM
n
✓�Fn
T
◆
g
n
=: exp
S
n
k
B
!,
F
n
:= En
� T · Sn
%
Mario Angelelli Geometric structures in statistical physics
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limit in statistical physics
Our starting point is
kB
! 0. (12)
In more details,
Definition
Given a physical statistical system specified by a partition function and an associatedfree energy, its tropical limit is defined as the simultaneous limit for k
B
and Avogadronumber N
A
such that their product is kept constant, i.e.
kB
! 0+, NA
! 1 : kB
· NA
=: R is constant.
F = �kB
T lnX
n
gn
exp
✓� EnkB
T
◆�! F
trop
(T ) = �TM
n
✓�Fn
T
◆
g
n
=: exp
S
n
k
B
!,
F
n
:= En
� T · Sn
%
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limit in statistical physics
Our starting point is
kB
! 0. (12)
In more details,
Definition
Given a physical statistical system specified by a partition function and an associatedfree energy, its tropical limit is defined as the simultaneous limit for k
B
and Avogadronumber N
A
such that their product is kept constant, i.e.
kB
! 0+, NA
! 1 : kB
· NA
=: R is constant.
F = �kB
T lnX
n
gn
exp
✓� EnkB
T
◆�! F
trop
(T ) = �TM
n
✓�Fn
T
◆
g
n
=: exp
S
n
k
B
!,
F
n
:= En
� T · Sn
%
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IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
On the basis of (2), particular physical phenomena can be studied in a new perspective[A. & Konopelchenko, 2015].
tropical dictionary for thermodynamic and statistical quantities, e.g. tropical freeenergy and tropical Gibbs probability distribution. It provides a simplecombinatorial framework to investigate the system.
Non-trivial phenomena come out in case of exponential degenerations gn
ofenergy levels through the reparametrization
gn
:= exp
✓Sn
kB
◆. (13)
Many phenomena related to exponential degenerations: residual entropy [Pauling,1935], topological order [Wen, 2013], frustrated systems ( [Diep, 2005; Lieb & Wu,1972], mainly spin ices [Gingras, 2011] and spin glasses [Parisi et al., 1987]), HEP(QCD [Blanchard et al., 2004]), cosmology [Carlip, 1997]. Some of these (residualentropy, limiting temperature, ultrametricity) are discussed in the tropical algebraformalism.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
On the basis of (2), particular physical phenomena can be studied in a new perspective[A. & Konopelchenko, 2015].
tropical dictionary for thermodynamic and statistical quantities, e.g. tropical freeenergy and tropical Gibbs probability distribution. It provides a simplecombinatorial framework to investigate the system.
Non-trivial phenomena come out in case of exponential degenerations gn
ofenergy levels through the reparametrization
gn
:= exp
✓Sn
kB
◆. (13)
Many phenomena related to exponential degenerations: residual entropy [Pauling,1935], topological order [Wen, 2013], frustrated systems ( [Diep, 2005; Lieb & Wu,1972], mainly spin ices [Gingras, 2011] and spin glasses [Parisi et al., 1987]), HEP(QCD [Blanchard et al., 2004]), cosmology [Carlip, 1997]. Some of these (residualentropy, limiting temperature, ultrametricity) are discussed in the tropical algebraformalism.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
On the basis of (2), particular physical phenomena can be studied in a new perspective[A. & Konopelchenko, 2015].
tropical dictionary for thermodynamic and statistical quantities, e.g. tropical freeenergy and tropical Gibbs probability distribution. It provides a simplecombinatorial framework to investigate the system.
Non-trivial phenomena come out in case of exponential degenerations gn
ofenergy levels through the reparametrization
gn
:= exp
✓Sn
kB
◆. (13)
Many phenomena related to exponential degenerations: residual entropy [Pauling,1935], topological order [Wen, 2013], frustrated systems ( [Diep, 2005; Lieb & Wu,1972], mainly spin ices [Gingras, 2011] and spin glasses [Parisi et al., 1987]), HEP(QCD [Blanchard et al., 2004]), cosmology [Carlip, 1997]. Some of these (residualentropy, limiting temperature, ultrametricity) are discussed in the tropical algebraformalism.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
On the basis of (2), particular physical phenomena can be studied in a new perspective[A. & Konopelchenko, 2015].
tropical dictionary for thermodynamic and statistical quantities, e.g. tropical freeenergy and tropical Gibbs probability distribution. It provides a simplecombinatorial framework to investigate the system.
Non-trivial phenomena come out in case of exponential degenerations gn
ofenergy levels through the reparametrization
gn
:= exp
✓Sn
kB
◆. (13)
Many phenomena related to exponential degenerations: residual entropy [Pauling,1935], topological order [Wen, 2013], frustrated systems ( [Diep, 2005; Lieb & Wu,1972], mainly spin ices [Gingras, 2011] and spin glasses [Parisi et al., 1987]), HEP(QCD [Blanchard et al., 2004]), cosmology [Carlip, 1997]. Some of these (residualentropy, limiting temperature, ultrametricity) are discussed in the tropical algebraformalism.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
These features are unseen in previous approach to tropical limit as a low temperaturelimit [Marcolli & Thorngren, 2012; Kapranov, 2011].
conceptual di↵erence: T ! 0 describes a boundary of the phase space, kB
! 0 is adi↵erent statistics which filters some aspects to highlight other ones;
preserves thermodynamic relations leaving T as a free parameter;
freedom in varying temperature can be used to study previous phenomena in greatgenerality including negative and limiting temperatures [Rumer, 1960; Atick &Witten, 1988]. ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
These features are unseen in previous approach to tropical limit as a low temperaturelimit [Marcolli & Thorngren, 2012; Kapranov, 2011].
conceptual di↵erence: T ! 0 describes a boundary of the phase space, kB
! 0 is adi↵erent statistics which filters some aspects to highlight other ones;
preserves thermodynamic relations leaving T as a free parameter;
freedom in varying temperature can be used to study previous phenomena in greatgenerality including negative and limiting temperatures [Rumer, 1960; Atick &Witten, 1988]. ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
These features are unseen in previous approach to tropical limit as a low temperaturelimit [Marcolli & Thorngren, 2012; Kapranov, 2011].
conceptual di↵erence: T ! 0 describes a boundary of the phase space, kB
! 0 is adi↵erent statistics which filters some aspects to highlight other ones;
preserves thermodynamic relations leaving T as a free parameter;
freedom in varying temperature can be used to study previous phenomena in greatgenerality including negative and limiting temperatures [Rumer, 1960; Atick &Witten, 1988]. ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Motivations
These features are unseen in previous approach to tropical limit as a low temperaturelimit [Marcolli & Thorngren, 2012; Kapranov, 2011].
conceptual di↵erence: T ! 0 describes a boundary of the phase space, kB
! 0 is adi↵erent statistics which filters some aspects to highlight other ones;
preserves thermodynamic relations leaving T as a free parameter;
freedom in varying temperature can be used to study previous phenomena in greatgenerality including negative and limiting temperatures [Rumer, 1960; Atick &Witten, 1988]. ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some results: thermodynamics
Etrop
:= limk
B
!0E , F
trop
:= limk
B
!0F and w
n,trop = limk
B
!0(k
B
· lnwn
) . . .
,
Ftrop
(T ) = �TM
n
✓�Fn
T
◆(14)
(= min{F1,F2, ..,Fn, ..} if T > 0).Gibbs’ distribution:
wn
=
exp
✓� EnkT
◆
X
m
gm
exp
✓�EmkT
◆ wn,trop = �Sn + Ftrop � Fn
T. (15)
Tropical Gibbs’ distribution for levels is
Wn,trop = wn,trop + Sn =
Ftrop
� Fn
T(16)
which satisfies the normalization conditionM
n
Wn,trop = 0. (17)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some results: thermodynamics
Etrop
:= limk
B
!0E , F
trop
:= limk
B
!0F and w
n,trop = limk
B
!0(k
B
· lnwn
) . . . ,
Ftrop
(T ) = �TM
n
✓�Fn
T
◆(14)
(= min{F1,F2, ..,Fn, ..} if T > 0).Gibbs’ distribution:
wn
=
exp
✓� EnkT
◆
X
m
gm
exp
✓�EmkT
◆ wn,trop = �Sn + Ftrop � Fn
T. (15)
Tropical Gibbs’ distribution for levels is
Wn,trop = wn,trop + Sn =
Ftrop
� Fn
T(16)
which satisfies the normalization conditionM
n
Wn,trop = 0. (17)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some results: thermodynamics
Etrop
:= limk
B
!0E , F
trop
:= limk
B
!0F and w
n,trop = limk
B
!0(k
B
· lnwn
) . . . ,
Ftrop
(T ) = �TM
n
✓�Fn
T
◆(14)
(= min{F1,F2, ..,Fn, ..} if T > 0).Gibbs’ distribution:
wn
=
exp
✓� EnkT
◆
X
m
gm
exp
✓�EmkT
◆ wn,trop = �Sn + Ftrop � Fn
T. (15)
Tropical Gibbs’ distribution for levels is
Wn,trop = wn,trop + Sn =
Ftrop
� Fn
T(16)
which satisfies the normalization conditionM
n
Wn,trop = 0. (17)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some results: thermodynamics
Etrop
:= limk
B
!0E , F
trop
:= limk
B
!0F and w
n,trop = limk
B
!0(k
B
· lnwn
) . . . ,
Ftrop
(T ) = �TM
n
✓�Fn
T
◆(14)
(= min{F1,F2, ..,Fn, ..} if T > 0).Gibbs’ distribution:
wn
=
exp
✓� EnkT
◆
X
m
gm
exp
✓�EmkT
◆ wn,trop = �Sn + Ftrop � Fn
T. (15)
Tropical Gibbs’ distribution for levels is
Wn,trop = wn,trop + Sn =
Ftrop
� Fn
T(16)
which satisfies the normalization conditionM
n
Wn,trop = 0. (17)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Singular locus
Singular locus: coincidence of at least two dominant terms Fi
= Fj
� Fk
,i 6= j 6= k 6= i , arising for certain values of intensive parameter. One has singular points(i.e. transition temperatures)
T ⇤ =En0 � Em0
Sn0 � Sm0
(18)
which are related by
(Si
� Sk
)T ⇤ik
+ (Sk
� Sl
)T ⇤kl
+ (Sl
� Si
)T ⇤li
= 0, i 6= k 6= l 6= i , i , k, l = 1, ..., n. (19)Near the singular temperature T ⇤ it is �W
trop
⇠ (T ⇤ � T )
�Wtrop
= Wn0,trop(T )�Wm0,trop(T ) = Fm0 � Fn0T = �Strop ·
✓T ⇤
T� 1◆
(20)
and non-dominant terms become relevant at T ⇤: expanding
lnwn
=w
n,trop
kB
+ ln w̃n
+ ..., (21)
#{n : Fn
= Ftrop
} = m , lnwn
= � Snk
B
� Fn(T⇤)�Fn0 (T⇤)kT
⇤ � lnm +O(kB).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Singular locus
Singular locus: coincidence of at least two dominant terms Fi
= Fj
� Fk
,i 6= j 6= k 6= i , arising for certain values of intensive parameter. One has singular points(i.e. transition temperatures)
T ⇤ =En0 � Em0
Sn0 � Sm0
(18)
which are related by
(Si
� Sk
)T ⇤ik
+ (Sk
� Sl
)T ⇤kl
+ (Sl
� Si
)T ⇤li
= 0, i 6= k 6= l 6= i , i , k, l = 1, ..., n. (19)
Near the singular temperature T ⇤ it is �Wtrop
⇠ (T ⇤ � T )
�Wtrop
= Wn0,trop(T )�Wm0,trop(T ) = Fm0 � Fn0T = �Strop ·
✓T ⇤
T� 1◆
(20)
and non-dominant terms become relevant at T ⇤: expanding
lnwn
=w
n,trop
kB
+ ln w̃n
+ ..., (21)
#{n : Fn
= Ftrop
} = m , lnwn
= � Snk
B
� Fn(T⇤)�Fn0 (T⇤)kT
⇤ � lnm +O(kB).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Singular locus
Singular locus: coincidence of at least two dominant terms Fi
= Fj
� Fk
,i 6= j 6= k 6= i , arising for certain values of intensive parameter. One has singular points(i.e. transition temperatures)
T ⇤ =En0 � Em0
Sn0 � Sm0
(18)
which are related by
(Si
� Sk
)T ⇤ik
+ (Sk
� Sl
)T ⇤kl
+ (Sl
� Si
)T ⇤li
= 0, i 6= k 6= l 6= i , i , k, l = 1, ..., n. (19)Near the singular temperature T ⇤ it is �W
trop
⇠ (T ⇤ � T )
�Wtrop
= Wn0,trop(T )�Wm0,trop(T ) = Fm0 � Fn0T = �Strop ·
✓T ⇤
T� 1◆
(20)
and non-dominant terms become relevant at T ⇤: expanding
lnwn
=w
n,trop
kB
+ ln w̃n
+ ..., (21)
#{n : Fn
= Ftrop
} = m , lnwn
= � Snk
B
� Fn(T⇤)�Fn0 (T⇤)kT
⇤ � lnm +O(kB).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Singular locus
Singular locus: coincidence of at least two dominant terms Fi
= Fj
� Fk
,i 6= j 6= k 6= i , arising for certain values of intensive parameter. One has singular points(i.e. transition temperatures)
T ⇤ =En0 � Em0
Sn0 � Sm0
(18)
which are related by
(Si
� Sk
)T ⇤ik
+ (Sk
� Sl
)T ⇤kl
+ (Sl
� Si
)T ⇤li
= 0, i 6= k 6= l 6= i , i , k, l = 1, ..., n. (19)Near the singular temperature T ⇤ it is �W
trop
⇠ (T ⇤ � T )
�Wtrop
= Wn0,trop(T )�Wm0,trop(T ) = Fm0 � Fn0T = �Strop ·
✓T ⇤
T� 1◆
(20)
and non-dominant terms become relevant at T ⇤: expanding
lnwn
=w
n,trop
kB
+ ln w̃n
+ ..., (21)
#{n : Fn
= Ftrop
} = m , lnwn
= � Snk
B
� Fn(T⇤)�Fn0 (T⇤)kT
⇤ � lnm +O(kB).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Nested tropical limit
The function e� 1
k
B is not analytic around kB
= 0, thus standard perturbative tools (e.g.,series expansion) may fail.
Proposition
Zeroth and first order corrections of F = F (kB
) correspond to data Ftrop
and m as in(21), respectively. The `-th order coe�cient in the Taylor expansion of F near k
B
= 0vanishes, for all ` � 2.However, one can use perturbation theory and additional structures, e.g. the nestingform of the P.F. to apply the tropical limit at di↵erent levels ?
Z(x) =eµ0(x) ·⇣⌫0 + e
µ1(x)�µ0(x) ·⇣⌫1 + e
µ2(x)�µ1(x) · (⌫2 + . . .· · · ·
⇣⌫L�2 + e
µL�1(x)�µL�2(x)·
⇣vL�1 + e
µL
(x)�µL�1(x) · ⌫
L
⌘. . .⌘⌘⌘⌘
. (22)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Nested tropical limit
The function e� 1
k
B is not analytic around kB
= 0, thus standard perturbative tools (e.g.,series expansion) may fail.
Proposition
Zeroth and first order corrections of F = F (kB
) correspond to data Ftrop
and m as in(21), respectively. The `-th order coe�cient in the Taylor expansion of F near k
B
= 0vanishes, for all ` � 2.
However, one can use perturbation theory and additional structures, e.g. the nestingform of the P.F. to apply the tropical limit at di↵erent levels ?
Z(x) =eµ0(x) ·⇣⌫0 + e
µ1(x)�µ0(x) ·⇣⌫1 + e
µ2(x)�µ1(x) · (⌫2 + . . .· · · ·
⇣⌫L�2 + e
µL�1(x)�µL�2(x)·
⇣vL�1 + e
µL
(x)�µL�1(x) · ⌫
L
⌘. . .⌘⌘⌘⌘
. (22)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Nested tropical limit
The function e� 1
k
B is not analytic around kB
= 0, thus standard perturbative tools (e.g.,series expansion) may fail.
Proposition
Zeroth and first order corrections of F = F (kB
) correspond to data Ftrop
and m as in(21), respectively. The `-th order coe�cient in the Taylor expansion of F near k
B
= 0vanishes, for all ` � 2.However, one can use perturbation theory and additional structures, e.g. the nestingform of the P.F. to apply the tropical limit at di↵erent levels ?
Z(x) =eµ0(x) ·⇣⌫0 + e
µ1(x)�µ0(x) ·⇣⌫1 + e
µ2(x)�µ1(x) · (⌫2 + . . .· · · ·
⇣⌫L�2 + e
µL�1(x)�µL�2(x)·
⇣vL�1 + e
µL
(x)�µL�1(x) · ⌫
L
⌘. . .⌘⌘⌘⌘
. (22)
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some figures
Ex: 3-level case: classifying cases.
(a) Ftrop
in three level case,S1 < S2 < S3,T
⇤23 > T
⇤13 > T
⇤12.
(b) Ftrop
in three level case,S1 < S2 < S3,T
⇤23 < T
⇤13 < T
⇤12.
(c) Ftrop
in three level case,S3 < S1 < S2: at0 < T < T⇤12, Etrop = E1,S
trop
= S1.
What if the spectrum is {Ei
: i 2 N} or {Ei
: i 2 Z}? Singularity can occur whenmin{F1,F2, ...} does not exist! Even in Rmax = (R [ {�1},max,+), and hence�1 2 Rmax, Wn,trop does not define a distribution of probabilities obeying the tropicalnormalization condition (17). ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some figures
Ex: 3-level case: classifying cases.
(a) Ftrop
in three level case,S1 < S2 < S3,T
⇤23 > T
⇤13 > T
⇤12.
(b) Ftrop
in three level case,S1 < S2 < S3,T
⇤23 < T
⇤13 < T
⇤12.
(c) Ftrop
in three level case,S3 < S1 < S2: at0 < T < T⇤12, Etrop = E1,S
trop
= S1.
What if the spectrum is {Ei
: i 2 N} or {Ei
: i 2 Z}? Singularity can occur whenmin{F1,F2, ...} does not exist!
Even in Rmax = (R [ {�1},max,+), and hence�1 2 Rmax, Wn,trop does not define a distribution of probabilities obeying the tropicalnormalization condition (17). ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Some figures
Ex: 3-level case: classifying cases.
(a) Ftrop
in three level case,S1 < S2 < S3,T
⇤23 > T
⇤13 > T
⇤12.
(b) Ftrop
in three level case,S1 < S2 < S3,T
⇤23 < T
⇤13 < T
⇤12.
(c) Ftrop
in three level case,S3 < S1 < S2: at0 < T < T⇤12, Etrop = E1,S
trop
= S1.
What if the spectrum is {Ei
: i 2 N} or {Ei
: i 2 Z}? Singularity can occur whenmin{F1,F2, ...} does not exist! Even in Rmax = (R [ {�1},max,+), and hence�1 2 Rmax, Wn,trop does not define a distribution of probabilities obeying the tropicalnormalization condition (17). ?
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limits of statistical hypersurfaces and amoebas
More tropical limits for independent variables (parameters) or dependent ones(“microscopic free energies” f↵).
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limits of statistical hypersurfaces and amoebas
More tropical limits for independent variables (parameters) or dependent ones(“microscopic free energies” f↵).
Dependent variables: xi
= �x?i
for large values of variables xi
� 1, i = 1, . . . , n andx?i
finite.
Statistical hypersurface is piecewise smooth. On the singular locus geometriccharacteristics can be regularized.
For linear f↵:
xn+1 = �x
?n+1,
The metric is non-degenerate in the tropicallimit, the resulting tropical hypersurface ispiecewise-flat.
Nonlinear f↵ require a double scalingtropical limit:
xn+1 = �
d · x?n+1.
Highly degenerate (rank 1) metric.
Connected components of the complement of the tropical graph of the first kind(“slow variables”) are unbounded.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limits of statistical hypersurfaces and amoebas
More tropical limits for independent variables (parameters) or dependent ones(“microscopic free energies” f↵).
Dependent variables: xi
= �x?i
for large values of variables xi
� 1, i = 1, . . . , n andx?i
finite.
Statistical hypersurface is piecewise smooth. On the singular locus geometriccharacteristics can be regularized.
For linear f↵:
xn+1 = �x
?n+1,
The metric is non-degenerate in the tropicallimit, the resulting tropical hypersurface ispiecewise-flat.
Nonlinear f↵ require a double scalingtropical limit:
xn+1 = �
d · x?n+1.
Highly degenerate (rank 1) metric.
Connected components of the complement of the tropical graph of the first kind(“slow variables”) are unbounded.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limits of statistical hypersurfaces and amoebas
More tropical limits for independent variables (parameters) or dependent ones(“microscopic free energies” f↵).
Dependent variables: xi
= �x?i
for large values of variables xi
� 1, i = 1, . . . , n andx?i
finite.
Statistical hypersurface is piecewise smooth. On the singular locus geometriccharacteristics can be regularized.
For linear f↵:
xn+1 = �x
?n+1,
The metric is non-degenerate in the tropicallimit, the resulting tropical hypersurface ispiecewise-flat.
Nonlinear f↵ require a double scalingtropical limit:
xn+1 = �
d · x?n+1.
Highly degenerate (rank 1) metric.
Connected components of the complement of the tropical graph of the first kind(“slow variables”) are unbounded.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limits of statistical hypersurfaces and amoebas
More tropical limits for independent variables (parameters) or dependent ones(“microscopic free energies” f↵).
Dependent variables: xi
= �x?i
for large values of variables xi
� 1, i = 1, . . . , n andx?i
finite.
Statistical hypersurface is piecewise smooth. On the singular locus geometriccharacteristics can be regularized.
For linear f↵:
xn+1 = �x
?n+1,
The metric is non-degenerate in the tropicallimit, the resulting tropical hypersurface ispiecewise-flat.
Nonlinear f↵ require a double scalingtropical limit:
xn+1 = �
d · x?n+1.
Highly degenerate (rank 1) metric.
Connected components of the complement of the tropical graph of the first kind(“slow variables”) are unbounded.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limits of statistical hypersurfaces and amoebas
More tropical limits for independent variables (parameters) or dependent ones(“microscopic free energies” f↵).
Dependent variables: xi
= �x?i
for large values of variables xi
� 1, i = 1, . . . , n andx?i
finite.
Statistical hypersurface is piecewise smooth. On the singular locus geometriccharacteristics can be regularized.
For linear f↵:
xn+1 = �x
?n+1,
The metric is non-degenerate in the tropicallimit, the resulting tropical hypersurface ispiecewise-flat.
Nonlinear f↵ require a double scalingtropical limit:
xn+1 = �
d · x?n+1.
d > 1 is the highest degreeamong the homogeneous partsof the interactions {f↵}.
Highly degenerate (rank 1) metric.
Connected components of the complement of the tropical graph of the first kind(“slow variables”) are unbounded.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
Tropical limit in statistical physics
Tropical limits of statistical hypersurfaces and amoebas
More tropical limits for independent variables (parameters) or dependent ones(“microscopic free energies” f↵).
Dependent variables: xi
= �x?i
for large values of variables xi
� 1, i = 1, . . . , n andx?i
finite.
Statistical hypersurface is piecewise smooth. On the singular locus geometriccharacteristics can be regularized.
For linear f↵:
xn+1 = �x
?n+1,
The metric is non-degenerate in the tropicallimit, the resulting tropical hypersurface ispiecewise-flat.
Nonlinear f↵ require a double scalingtropical limit:
xn+1 = �
d · x?n+1.
Highly degenerate (rank 1) metric.
Connected components of the complement of the tropical graph of the first kind(“slow variables”) are unbounded.
Mario Angelelli Geometric structures in statistical physics
-
IntroductionGeometric micro-macro mappings
T