## Controlling non-equilibrium dynamics in lattice gas models

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2021-03-05##### Author

Mukhamadiarov, Ruslan Ilyich

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In recent years a new interesting research avenue has emerged in non-equilibrium statistical physics, namely studies of collective responses in spatially inhomogeneous systems. Whereas substantial progress has been made in understanding the origins and the often universal nature of cooperative behavior in systems far from equilibrium, it is still unclear whether it is possible to control their global collective stochastic dynamics through local manipulations. Therefore, a comprehensive characterization of spatially inhomogeneous non-equilibrium systems is required.
In the first system, we explore a variant of the Katz–Lebowitz–Spohn (KLS) driven lattice gas in two dimensions, where the lattice is split into two regions that are coupled to heat baths with distinct temperatures T > Tc and Tc respectively, where Tc indicates the critical temperature for phase ordering.
The geometry was arranged such that the temperature boundaries are oriented perpendicular or parallel to the external particle drive and resulting net current.
For perpendicular orientation of the temperature boundaries, in the hotter region, the system behaves like the (totally) asymmetric exclusion processes (TASEP), and experiences particle blockage in front of the interface to the critical region.
This blockage is induced by extended particle clusters, growing logarithmically with system size, in the critical region.
We observe the density profiles in both high- and low-temperature subsystems to be similar to the well-characterized coexistence and maximal-current phases in (T)ASEP models with open boundary conditions, which are respectively governed by hyperbolic and trigonometric tangent functions.
Yet if the lower temperature is set to Tc, we detect marked fluctuation corrections to the mean-field density profiles, e.g., the corresponding critical KLS power-law density decay near the interfaces into the cooler region.
For parallel orientation of the temperature boundaries, we have explored the changes in the dynamical behavior of the hybrid KLS model that are induced by our choice of the hopping rates across the temperature boundaries.
If these hopping rates at the interfaces satisfy particle-hole symmetry, the current difference across them generates a vector flow diagram akin to an infinite flat vortex sheet.
We have studied the finite-size scaling of the particle density fluctuations in both temperature regions, and observed that it is controlled by the respective temperature values. If the colder subsystem is maintained at the KLS critical temperature, while the hotter subsystem's temperature is set much higher, the interface current greatly suppresses particle exchange between the two regions. As a result of the ensuing effective subsystem decoupling, strong fluctuations persist in the critical region, whence the particle density fluctuations scale with the KLS critical exponents. However, if both temperatures are set well above the critical temperature, the particle density fluctuations scale according to the totally asymmetric exclusion process. We have also measured the entropy production rate in both subsystems; it displays intriguing algebraic decay in the critical region, while it saturates quickly at a small but non-zero level in the hotter region.
The second system is a lattice gas that simulates the spread of COVID-19 epidemics using the paradigmatic stochastic Susceptible-Infectious-Recovered (SIR) model. In our effort to control the spread of the infection of a lattice, we robustly find that the intensity and spatial spread on the
epidemic recurrence wave can be limited to a manageable extent provided release of these restrictions is delayed sufficiently (for a duration of at least thrice the time until the peak of the unmitigated outbreak).

##### General Audience Abstract

In recent years a new interesting research avenue has emerged in far-from-equilibrium statistical physics, namely studies of collective behavior in spatially non-uniform systems. Whereas substantial progress has been made in understanding the origins and the often universal nature of cooperative behavior in systems far from equilibrium, it is still unclear whether it is possible to control their global collective and randomly determined dynamics through local manipulations. Therefore, a comprehensive characterization of spatially non-uniform systems out of equilibrium is required.
In the first system, we explore a variant of the two-dimensional lattice gas with completely biased diffusion in one direction and attractive particle interactions. By lattice gas we mean a lattice filled with particles that can hop on nearest-neighbor empty sites. The system we are considering is a lattice that is split into two regions, which in turn are maintained at distinct temperatures T > Tc and Tc, respectively, with Tc indicating the critical temperature for the second-order phase transition.
The geometry of the lattice was arranged such that the temperature boundaries are oriented perpendicular or parallel to the external particle drive that is responsible for a completely biased diffusion.
When the temperature boundaries are oriented perpendicular to the drive, in the hotter region with temperature T > Tc, the system evolves as if there are no attractive interactions between the particles, and experiences particle blockage in front of the temperature boundary from the hotter region held at T>Tc to the critical region held at Tc.
This accumulation of particles at the temperature boundary is induced by elongated collections of particle, i.e., particle clusters in the critical region.
We observe the particle density profiles (density(x) vs x plots) in both high-and low-temperature subsystems to be similar to the density profiles found for other well-characterized (T)ASEP models with open boundary conditions, which are in the coexistence and maximal-current phases, and which are respectively governed by hyperbolic and trigonometric tangent functions.
Yet if the lower temperature is set to Tc, we detect marked corrections to the hyperbolic and trigonometric tangent-like density profiles due to fluctuations, e.g., we observe the algebraic power-law decay of the density near the interfaces into the cooler region with the critical KLS exponent.
For a parallel orientation of the temperature boundaries, we have explored the changes in the particle dynamics of the two-temperature KLS model that are induced by our choice of the particle hopping rates across the temperature boundaries.
If these particle hopping rates at the temperature interfaces satisfy particle-hole symmetry (i.e. remain unchanged when particles are replaced with holes and vice versa), the particle current difference across them generates a current vector flow diagram akin to an infinite flat vortex sheet.
We have studied how the particle density fluctuations in both temperature regions scale with the system size, and observed that the scaling is controlled by the respective temperature values. If the colder subsystem is maintained at the KLS critical temperature Tcold = Tc, while the hotter subsystem's temperature is set much higher Thot >> Tc, the particle currents at the interface greatly suppresses particle exchange between the two temperature regions. As a result of the ensuing effective subsystem separation from each other, strong fluctuations persist in the critical region, whence the particle density fluctuations scale with the KLS critical exponents. However, if both temperatures are set well above the critical temperature, the particle density fluctuations scale with different scaling exponents, that fall into the totally asymmetric exclusion process (TASEP) universality class. We have also measured the rate of the entropy production in both subsystems; it displays intriguing algebraic decay in the critical region, while it reaches quickly a small but non-zero value in the hotter region.
The second system is a lattice filled with particles of different types that hop around the lattice and are subjected to different sorts of reactions. That process simulates the spread of the COVID-19 epidemic using the paradigmatic random-process-based Susceptible-Infectious-Recovered (SIR) model. In our effort to control the spread of the infection of a lattice, we robustly find that the intensity and spatial spread of the
epidemic second wave can be limited to a manageable extent provided release of these restrictions is delayed sufficiently (for a duration of at least thrice the time until the peak of the unmitigated outbreak).

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- Doctoral Dissertations [14966]

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