Non-equilibrium Statistical Mechanics

Although much is known about systems at equilibrium, their non-equilibrium counterparts remain poorly understood.  For example, even the humble one-dimensional Ising model exhibits a rich range of behaviors when it is driven far from equilibrium.  These far-from-equilibrium states are difficult to describe; there is no existing conceptual framework of the same power and breadth as the one developed for equilibrium systems by Boltzmann, Gibbs, and others.

I am interested in one of the most basic ways of driving a system: setting different pieces of the system at different temperatures.  I'm mostly interested in Ising-like models whose states are characterized by a spin configuration \(\{ \sigma_i \}\), where \(\sigma_i = \pm 1\) at some lattice sites \(i\).  Perhaps the most basic starting point for studying such systems is trying to solve for the probability \(P(\{ \sigma_i \},t)\) of observing a particular spin configuration \(\{ \sigma_i \}\) at time \(t\), and finding steady-state solutions \(P^*( \{ \sigma_i \})\) as \(t \rightarrow \infty\).  The probability obeys a conversation law called the master equation:
$$
\partial_t P(\{ \sigma_i \},t) = \sum_{\{ \sigma_i'\}} \left[\omega( \{ \sigma_i' \} \rightarrow \{ \sigma_i \})P(\{ \sigma_i' \},t)- \omega(\{ \sigma_i \} \rightarrow \{ \sigma_i' \})P(\{ \sigma_i \},t) \right],
$$ where the \(\omega\)'s are probability rates of moving from one spin configuration to another one.  Note that the sum here is over all possible spin configurations \(\{ \sigma_i' \}\).  In general, it is very difficult to solve the master equation.  However, in certain cases, a judicious choice of \(\omega\) allows us to make progress.





We considered a one-dimensional Ising chain driven by Glauber dynamics [1].  In this case, the \(\omega\)'s are non-zero just for configurations  \(\{ \sigma_i \}\) and  \(\{ \sigma_i' \}\) that differ by a single spin flip.  If we identify the spin flip as  \(\sigma_x\), then we may replace the transition  \(\{ \sigma_i\} \rightarrow \{ \sigma_i' \}\) by just the value of the spin  \(\sigma_x\) at site  \(x\) in the initial configuration.  Our rates are:
$$
\omega(\sigma_x) = \frac{1}{2 \Delta t} \left[ 1 - \frac{\gamma(x)}{2} \, \sigma_x (\sigma_{x-1}+\sigma_{x+1}) \right],
$$ where  \(\gamma(x) = \tanh (2 \beta_x J)\), where  \(J\) is the using Ising coupling strength and  \(\beta_x=(k_B T_x)^{-1}\) is the inverse temperature of the spin at site  \(x\).  We were able to solve for various quantities such as the energy flux $F(x)$ through the system for a system in which  \(T_x = \infty\) for all  \(x \leq 0\) and  \(T_x = T_c\) for all  \(x > 0\) [2,3]. Such analytic results pave the way for more general understanding of such systems.

Our group has recently started working on a driven system in two and three dimensions consisting of a binary mixture of particles. These particles have simple excluded volume interactions and, in addition, particles of opposite types cannot occupy nearest neighbor locations. When the particles are subjected to a drive, amazing striped patterns emerge [4].



References

[1] R. J. Glauber Time-dependent statistics of the Ising model, Journal of Mathematical Physics 4, 294 (1963)
[2] M. O. Lavrentovich and R. K. P. Zia Energy flux near the junction of two Ising chains at different temperatures, EPL 91(5) 50003 (2010)
[3] M. O. Lavrentovich Steady-state properties of coupled hot and cold Ising chains, Journal of Physics A: Mathematical and Theoretical 45 085002 (2012)
[4] R. Dickman and R. K. P. Zia Driven Widom-Rowlinson lattice gas Physical Review E 97 062126 (2018)