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

[2] M. O. Lavrentovich and R. K. P. Zia

[3] M. O. Lavrentovich

[4] R. Dickman and R. K. P. Zia *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)*Driven Widom-Rowlinson lattice gas*Physical Review E

**97**062126 (2018)