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