stochastic theory of nonequilibrium statistical mechanicsbicmr.pku.edu.cn/~gehao/chinese...
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Stochastic theory of nonequilibrium statistical mechanics
1Beijing International Center for Mathematical Research2Biodynamic Optical Imaging Center
Peking University, Chinahttp://bicmr.pku.edu.cn/~gehao/
Ge, H.: Stochastic Theory of Nonequilibrium Statistical Physics (review). Advances in Mathematics(China) 43, 161-174 (2014)
Summary of Ge group
Stochastic Biophysics(Biomath)
Stochastic theory of nonequilibrium statistical mechanics
JSP06,08PRE09,10,13,14JCP12; JSTAT15
Nonequilibrium landscape theory and rate formulas
PRL09,15JRSI11; Chaos12
Stochastic modeling of biophysical systems
Statistical machine learning of single-cell data
JPCB08,13; JPA12Phys. Rep. 12Science13;Cell14;MSB15
Which kind of physical/chemical processes can be described by stochastic processes?
• Mesoscopic scale (time and space)
• Single-molecule and single-cell (subcellular) dynamics
• Trajectory perspective
Stochastic single-molecule and single-cell kinetics
Lu, et al. Science (1998)
Single-molecule enzyme kinetics
Eldar, A. and Elowitz, M. Nature (2010)
Choi, et al. Science (2008)
Single-cell kinetics
The fundamental equation in nonequilibrium thermodynamics
Ilya Prigogine(1917-2003)Nobel Prize
in 1977
Entropy production
SdSddS ie
0 k
kki XJSdepr
Clausius inequalityTQdS
Rudolf Clausius(1822-1888)
Second law of thermodynamics
0TQdSepr
rewrite
More general
Carl Eckart(1902-1973)
P.W. Bridgman(1882-1961)Nobel Prize
in 1946
Mathematical theory of nonequilibrium steady state
Min Qian (1927-)Recipient of Hua Loo-Keng Mathematics Prize (华罗庚数学奖) in 2013
Time-independent(stationary) Markov process
Reversible Michaelis-Menten enzyme kinetics
Kinetic scheme of a simple reversible enzyme. From the perspective of a single enzyme molecule, the reaction is unimolecular and cyclic.
Ge, H., Qian, M. and Qian, H.: Phys. Rep. (2012)
Ge, H.: J. Phys. Chem. B (2008)
Two reversibleMichaelis-Menten reactions
Min, et al. Nano Lett, (2005)
PS
Simulated turnover traces of a single molecule
Ge, H.: J. Phys. Chem. B (2008)
)(t
PSt :)( SPt :)( the number of occurrences of forward and backward cycles up to time t
Steady-state cycle fluxes and nonequilibrium steady state
.][][1
][][)(lim ssss
mPmS
mPP
mSS
t
ss JJ
KP
KS
KPV
KSV
ttJ
.][][1
][)(lim ;][][1
][)(lim
mPmS
mPP
t
ss
mPmS
mSS
t
ss
KP
KS
KPV
ttJ
KP
KS
KSV
ttJ
Michaelis-Menten kinetics
Chemical potential difference: )ln(][
][ln 0321
3201
ss
ss
BB JJTk
PkkkSkkkTk
Ge, H.: JPCB (2008)Ge, H., Qian, M. and Qian,H.: Phys. Rep. (2012)
00
ssJ
Waiting cycle times
ESE EP E ES EP E-2-3 -1 0 1 2 3 PS
k2k1 k3 k1 k2 k3
k-1 k-2 k-3 k-1 k-2 k-3
The kinetic scheme for computing the waiting cycle times, which also serves for molecular motors.
.1;1;1ssssssss J
TJ
TJJ
T
Generalized Haldane equality
).|()|( TTtTPTTtTP EE
Waiting cycle time T is independent of whether the enzyme completes a forward cycle or a backward cycle, although the probability weight of these two cycles might be rather different.
Carter, N. J.; Cross, R. A. Nature (2005)
Superposed distributions!
Average time course of forward and backward steps Ge, H.: J. Phys. Chem. B (2008); J. Phys. A (2012)
Jia, et al. submitted (2015): general Markovian jumping process
Ge, et al. submitted (2015): diffusion on a circle
Nonequilibrium steady state
Ge, H., et al. Phys. Rep. (2012)
321
321BPS kkk
kkklogTk
)|()|( TTtTPTTtTP EE
Ge, H. J. Phys. Chem. B (2008); J. Phys. A (2012)
Generalized Haldane Equality
Fluctuation theorem of fluctuating chemical work
Tkn
E
E BentWP
ntWP
))(())((
1)(
TktW
BeSecond law in terms of equality
Free energy conservation0
kkkkkklogTkepr
321
321B
Entropy production: Free energy dissipationFree energy input
PS 0.mEquilibriu
Traditional Second law
ttW
Transient fluctuation theorem for cyclic chemical work
2
'1
l2
l12
'1
l2
2'
1l12
'1
ltv,ltvPltv,ltvP
ltv,ltvPltv,ltvP212
1
tvtv,
'21
21etg
tgtg
tgtg
2211221
21121
log,loglog,
,log,
Ge, H.: JPA (2012)
tvtWtvtW '2211 log ,log
1eee tWtWtWtW 2121
Master equation model describing the system
No matter starting from any initial distribution, it will finally approach its stationary distribution satisfying
01
N
jij
ssiji
ssj kpkp
j
ijijiji kpkp
dttdp )(
Consider a motor protein with N different conformations R1,R2,…,RN. kij is the first-order or pseudo-first-order rate constants for the reaction Ri→Rj.
Self-assembly or self-organization
ijeqiji
eqj kpkp
Detailed balance (equilibrium state)
NESS thermodynamic force and entropy production rate
0 ji
ssij
ssij
ness AJeprT
jissjij
ssi
ssij kpkpJ
jissj
ijssi
Bssij kp
kplogTkA NESS thermodynamic force
NESS flux
NESS entropy production rate
Energy transduction efficiency at NESS
A mechanical system coupled fully reversibly to a chemical reactions, with a constant force resisting the mechanical movement driven by the chemical gradient.
l)(MechanicaOutput Energy (Chemical)Input Energy nessdh
1outputEnergy
outputEnergy inputEnergy outputEnergy
nessdh
or
(Chemical)Output Energy l)(MechanicaInput Energy nessdh
nessp
nessd Teh all dissipated! Ge, H.: PRE (2013)
Time-dependent case
j
ijijiji tkptkp
dttdp
Free energy j
kTtEB
jeTktF /)(log)(
j
kTtE
kTtEeqi
ssi j
i
eetptp /)(
/)(
)()(Boltzmann’s law
If {kij(t)} satisfys the detailed balance condition
01
N
jij
ssiji
ssj tktptktpQuasi-stationary distribution
Mathematical equivalence of Jarzynski and Hatano-Sasa equalities
Ge, H. and Qian, M., JMP (2007); Ge, H. and Jiang, D.Q., JSP (2008);
Jarzynski equality: local equilibrium
kTF
pp
kTW ee eq
/
)0()0(
/
.)0()0(
FW eqpp
Hatano-Sasa equality: without local equilibrium
1)0()0(
/
ss
ex
pp
kTQe
.kT/Q
S
ex
pp ss
00
Same theorem for time-dependent Markov process
Are these inequalities already known in the Second Law of classic thermodynamics? Do they only hold for the whole transition process between two steady states?
The traditional Clausius inequality can be in a differential form.
Decomposition of mesoscopic thermodynamic forces
tAtA
tktptktp
TktA ijssij
jij
ijiBij log
i j
0 ji
ijijp tAtJteT Entropy production
0 ji
ssijijhk tAtJtQ )()()(Housekeeping heat
Free energy dissipation 0 ji
ijijd tAtJtf )()()(
t t t dhkp fQeT Ge, H., PRE (2009);
Ge, H. and Qian, H., PRE (2010) (2013)
tktp
tktpTktA
jissj
ijssi
Bssij log
0tep for any time t In the absence of external energy input and at steady state.
0tQhk for any time t In the absence of external energy input
0tfd for any time t At steady state
Two origins of nonequilibrium
Ge, H., PRE (2009); Ge, H. and Qian, H., PRE (2010) (2013)
0 ptot e
TQ
dtdS
The new Clausius inequality is stronger than the traditional one.
0 d
hktotex fT
QQT
QdtdS
Decomposition of entropy production rate
.QfeT,Q,f
hkdp
hkd
000
Ge, H., PRE (2009);
Ge, H. and Qian, H., PRE (2010) (2013)
TQe
dtdS tot
p
Two different definitions of entropy production rates
dxdvlogmxDFF
xDk
Ts:rPTs:Plog
TlimkEPR
ttvB
PsTts
st
TB
2
01
222
001
v,xFF
Spinney, R.E., and Ford, I.J., PRL (2012); Lee, H.K., Kwon, C., and Park, H., PRL (2013)
dxdvlog
mxDvx
xDkEPR ttvB
2
2 22
Kim, K.H. and Qian, H., PRL (2004)
correspond to time-reversibility and Maxwell-Boltzmann distribution for thermodynamic equilibrium respectively
00 21 EPR,EPR
thermodynamic equilibrium Ge, H.: PRE (2014)
When the external forceis only dependent on position
xGvxv,xF xxTk
xD
B
2
021 EPREPRthermodynamic equilibrium
xUxG
;TxT
x Thermal equilibrium
Mechanical equilibrium
Flow of kinetic energy along the spatial coordinate
t,xQt,xWJxEdtd kinetic
xxkinetict
00 qx JJEPR
xkinetict
kineticxq vxEJxJ (measurable) Heat flux
Ge, H.: PRE (2014)
Thermodynamic equivalence between mesoscopic and macroscopic scales
The entropy production rate in the small-noise limit
Celani, et al.: PRL (2012); Ge, H.: PRE (2014)
dxˆ
ˆJ
xTxEPREPR t
t
overxoverspatial
2
dxxˆxTx
xTknt
xB
over
22
62
Entropy production rate of the overdamped-limit
Anomalous contribution of EPR
Hence the overdamped approximation only keeps the dynamics rather than the second law of thermodynamics.
spatialEPREPRDecomposition of entropy production rate
Local reciprocal relation between linear coefficients
Ge, H.: JSTAT (2015)
qqqpqxq
qxqpxxoverx
XLXLxJXLXLJ
xˆxxTknL t
Bqq
32
68
xˆxxTL txx
x
xTkLL Bxqqx
2
Always hold, even for far-from-equilibrium system.
Reciprocal relation between Soret effect (thermal diffusion) and Dufour effect
Come from the second moment of velocity along each dimension.
Summary
Stochastic theory of nonequilibrium statistical mechanics is rigorously studied and further developed, including the cyclic dynamics with an unexpected equality, fluctuation theorems and decomposition of entropy production rate.
A stronger Clausius’ inequality emerges only in the nonequilibrium case;
Recently we are interested in the second-order stochastic system with temperature gradient.
Prof. Min QianPeking University
Acknowledgement
Prof. Hong QianUniversity of Washington
Fundings: NSFC, MOE of PRC