Maxim O. Lavrentovich: 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]. When the drive magnitude $ \delta $ is increased, the stripes narrow:

Although we largely do not understand this phenomenon, some progress has been made in understanding how such patterns emerge at low particle densities [5].

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)

[5] M. O. Lavrentovich, R. Dickman, and R. K. P. Zia Microemulsions in the driven Widom-Rowlinson lattice gas 104, 064135 (2021)