Two Temperature Non-equilibrium Ising Model in 1DNick Borchers

Outline

• Background• Non-equilibrium vs. Equilibrium systems• Master equation and Detailed Balance• Ising model

• Preliminary Results• Model description• Dependence on external temperature• Dependence on infinite temperature region size• Dependence of lattice size

• Future pursuits• Configuration Space characteristics and trajectories• Localized quantities and subsystems• Applications to living systems

Non-equilibrium versus equilibrium

Non-EquilibriumEquilibrium

Equilibrium System Features:• Probability proportional to Boltzmann factor:

• ‘Time Reversal’ Symmetry, or ‘Detailed Balance’

/H kTe

P CZ

Steady-State vs. Equilibrium

• Equilibrium is a special case of steady-state in which there is no steady flux through the system. This requires an isolated system, which is an idealization which must be carefully constructed.

• There may be Non-Equilibrium Steady States (NESS), in which the inputs and outputs of the system are balanced, but there is a flux through the system. Simple examples[1]:

• Systems in a NESS a notable for the presence of generic long-range correlations even when the interactions are short-ranged

Detailed Balance ‘Detail’: Master Equation

• : Probability of finding system in state i at time step τ

• : Transition rate or probability from state j to i.

• Probability at time τ+1:

• Master Equation:

jiw

1 j j ii i j i j i i

j j i

P w P w P w P

1 1j ii i i i j i i

j i

P P P w P w P

j ii i j j i

j i

P w P w P

Detailed Balance ‘Detail’: Steady State

Assume the existence of a stationary distribution P*, i.e.

Then

Detailed Balance holds if:

* *1P P

0j ii i j j i

j i

P w P w P

j ii j j iw P w P

Role of Simple Models

Typical NESS of physical interest are analytically intractable. Thus we turn to simple models. The goal?

• Account for as many physical features as possible

• Simplifying enough such remaining amenable to analytical or numerical solution[1]

Ising Model

• Spins σi ε {±1} on a discrete lattice

• Nearest neighbor Hamiltonian with interaction constant Jij

• 1-D equilibrium case solved by Ernst Ising (1924). No phase transition.

• 2-D equilibrium model solved by Lars Onsager(1944). Phase transition at critical T.

Hamiltonian:

,ij i j

i j

H J

Simulating the Ising Model

• Monte-Carlo simulation

• Metropolis Algorithm: Set transition rates to give desired Boltzmann distribution.• Detailed Balance + Probability of microstate:

• Glauber Dynamics: Random spin flips (ferromagnetism)

• Kawasaki Dynamics: Spin exchange (binary alloys)

j ii j j iw P w P

/H kTe

P CZ

/j

H kTiij

we

w

Two Temperature Ising Models

• “Convection cells induced by spontaneous symmetry breaking” M. Pleimling, B. Schmittmann, R.K.P. Zia [2]

“Formation of non-equilibrium modulated phases under local energy input” L.Li, M. Pleimling [3]

1-D Two Temperature Ising Model

• 1-D lattice• Periodic boundaries (ring)• Kawasaki dynamics• Typically half-filled (M=0)• Two coupled temperatures• Ising Hamiltonian:

4 tunable parameters:• Lattice size L• Sub-lattice size s• Temperature TL

• Temperature Ts, typically infinite

sT

LT

L s

s

,i j

i j

H

Detailed Imbalance

• Following the Metropolis algorithm, and assuming two independent equilibrium probability distributions, we have the following rates:

• These rates would be appropriate if the two regions were isolated, or perhaps far from the edges. At the boundaries, there is a conflict.

• Since the rates are set assuming the Boltzmann distribution for states, detailed balance is broken for all states.

/ ,1 1,1 ;sH kTsw e / ,1LH kT

Lw e

j ii j j iw P w P

Characterization Quantities

sT

LT

L s

s

• Average Local Energy (ALE): Average energy for a single bond. Bond energy may be ±1.

• Average Local Magnetization (ALM): Average spin at single lattice site. Center set to +1.

• Local Histograms for Occupation Percentage: Histograms for the number of occupied sites within a sub-lattice.

Results:ALM dependence on TL

L= 80, s = 20, Ts = ∞

Results:Sub-lattice Occupation

L=100, s = 25

Results:Occupation, TL dependence

L=100, s = 25, Ts=∞

Results:S-lattice Occupation, s dependence

L=100, kTL = 1, Ts=∞

Results:S-lattice Occupation, s dependence

L=100, kTL = 1, Ts=∞

Results:ALE dependence on s

L = 80, kTL = 1, Ts = ∞

Results:S-lattice Occupation, L dependence

s=L/4, kT L= 1, Ts = ∞

Future Work: Obvious Extensions

• Improved simulation framework for:• Generating results• Visualizing data

• Complete phase diagram

• Most importantly, develop detailed physical understanding

Configuration Space Topology

• Can general topological features of the configuration space be determined without recourse to explicit construction?

• What could these features, if determined, tell us about the dynamics of the system? Kawasaki Dynamics: L=6

Configuration Space Trajectories

• The configuration space topology for equilibrium and non-equilibrium systems is identical. Edge weights differ.

• Can the trajectories through configuration space be characterized, and how does their nature affect system dynamics?

• Absorbing states and transient flights

Kawasaki Dynamics: L=6, TL=0

Configuration Space Trajectories

• The configuration space topology for equilibrium and non-equilibrium systems is identical. Edge weights differ.

• Can the trajectories through configuration space be characterized, and how does their nature affect system dynamics?

• Absorbing states and transient flights

Kawasaki Dynamics: L=6, s=2, TL=0, Ts=∞

Energy Level Graph and Trajectories

• Simplified Graph• Complicated edge

weights

Subsystems and localized quantities

• For an isolated system in equilibrium, statistical mechanics provides the definition of quantities such as Temperature and Entropy: lnBS k P

1 S

T U

• Can these quantities be calculated for subsystems of an isolated system? If calculated, would these quantities be useful?

System

Subsystem

Non-equilibrium Physics and Living Systems

On life: “It feeds on negative entropy” – Erwin Schrödinger[5]

• Use Non-equilibrium models and techniques to study the origin of fundamental features of living systems, e.g. metabolism, reproduction. In particular…

• Homeostasis: The regulation of internal environment to maintain a constant state.• Can subsystems with this property arise naturally within non-

equilibrium environments? What conditions and dynamics, such as natural feedbacks, are required for…• Spontaneous local entropy reduction• Local temperature islands

Summary

• Non-equilibrium statistical mechanics is relevant to the behavior of a myriad of real-world physical systems

• Simple models such the Ising model may be used to develop an understanding and intuition for these overwhelmingly complex real systems.

• A simple 1-D Ising model with two temperatures has been studied, and shows unexpected and, as yet, unexplained behavior.

• It is hoped that in understanding these phenomena, perhaps through the development of new means of configuration space analysis, will lead to an understanding of some fundamental properties of living systems.

References

[1] Chou T, Mallick K, Zia RKP. Non-equilibrium statistical mechanics: From a paradigmatic model to biological transport.

[2] Pleimling M, Schmittmann B, Zia RKP. Convection cells induced by spontaneous symmetry breaking. EPL 89, 50001

[3] Li L, Pleimling M. Formation of non-equilibrium modulated phases under local energy input.

[4] Landua D, Binder K. A guide to Monte-Carlo simulations in statistical physics. Second Edition. Cambridge: Cambridge University Press; 2005.

[5] McKay, C. What is life – and how do we search for it in other worlds? PLoS Biol 2(9): e302.

Thank You!

Questions??