### 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)