Two Temperature Non-equilibrium I sing Model in 1D

1. Two Temperature Non-equilibrium Ising Model in 1D Nick Borchers

2. 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

3. Non-equilibrium versus equilibrium Equilibrium System Features: • Probability proportional to Boltzmann factor: • ‘Time Reversal’ Symmetry, or ‘Detailed Balance’

4. Steady-State vs. Equilibrium • Equilibrium is a special case of steady-statein 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

5. Detailed Balance ‘Detail’: Master Equation • : Probability of finding system in state iat time step τ • : Transition rate or probability from state j to i. • Probability at time τ+1: • Master Equation:

6. Detailed Balance ‘Detail’: Steady State Assume the existence of a stationary distribution P*, i.e. Then Detailed Balance holds if:

7. 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]

8. Ising Model • Spins σiε {±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:

9. 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)

10. 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]

11. 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

12. 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.

13. Characterization Quantities • 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.

14. Results:ALM dependence on TL L= 80, s = 20, Ts= ∞

15. Results:Sub-lattice Occupation L=100, s = 25

16. Results:Occupation, TL dependence L=100, s = 25, Ts=∞

17. Results:S-lattice Occupation, s dependence L=100, kTL = 1, Ts=∞

18. Results:S-lattice Occupation, s dependence L=100, kTL = 1, Ts=∞

19. Results:ALE dependence on s L = 80, kTL = 1, Ts = ∞

20. Results:S-lattice Occupation, L dependence s=L/4, kTL= 1, Ts = ∞

21. Future Work: Obvious Extensions • Improved simulation framework for: • Generating results • Visualizing data • Complete phase diagram • Most importantly, develop detailed physical understanding

22. 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

23. 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

24. 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=∞

25. Energy Level Graph and Trajectories • Simplified Graph • Complicated edge weights

26. Subsystems and localized quantities • For an isolated system in equilibrium, statistical mechanics provides the definition of quantities such as Temperature and Entropy: • Can these quantities be calculated for subsystems of an isolated system? If calculated, would these quantities be useful?

27. 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

28. 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.

29. 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? PLoSBiol 2(9): e302. 

30. Thank You!Questions??