### 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 .  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 .

#### References

 R. J. Glauber Time-dependent statistics of the Ising model, Journal of Mathematical Physics 4, 294 (1963)
 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)
 M. O. Lavrentovich Steady-state properties of coupled hot and cold Ising chains, Journal of Physics A: Mathematical and Theoretical 45 085002 (2012)
 R. Dickman and R. K. P. Zia Driven Widom-Rowlinson lattice gas Physical Review E 97 062126 (2018)