Phase transitions in nonequilibrium systems

Abstract

Mixed order phase transitions are transitions which have common features with both first order and second order transitions. I review some results obtained in the context of one of the prototypical models of mixed order transitions, the one-dimensional Ising model with long-range coupling that decays as truncated inverse square distance between spins. The correspondence between this model and the Poland Scheraga model of DNA denaturation, a subject to which Michael Fisher made substantial contributions, is then outlined. I. JOINING MICHAEL FISHER AS A POSTDOC AT CORNELLI was a postdoctoral fellow of Michael Fisher at Cornell in the years 1975-1977. Before arriving in Cornell I was a research fellow at Brookhaven National Lab where I applied symmetry considerations of Landau theory to argue that n-vector spin models with n ≥ 4 are in fact realized in a large variety of physical systems. Renormalization group analysis of these models thus provides their classification into universality classes. Furthermore, it suggests that when an accessible stable fixed point is lacking they should display a first order transition.At the time Michael was skeptical about the use of Landau theory, beyond it being perhaps a useful but uncontrolled approximation. So upon my arrival in Cornell he asked me to present my work in one of his famous Tuesday seminars. The seminar turned into a tutorial series of three lengthy and grueling lectures where Michael kept posing questions and making infinitely many comments and suggestions. After the third seminar he asked me to his office for a chat in which at some point he said, you know Landau theory is under appreciated in the West, and then he added candidly "...and maybe I am to blame". We ended up collaborating on a number of studies combining Landau's symmetry considerations with renormalization group calculations together with my old friend and his then student Eytan Domany. We have, for example, suggested a physical realization of the three state Potts model [1] and demonstrated how a symmetry breaking field can turn a first order transition into a continuous one.Cornell at the time was an exciting and intelectually stimulating place to be, and working with Michael was a unique experience. I have learned a lot from Michael's non-compromising approach to science, how to try to select and pursue meaningful scientific problems and how to conduct research. My stay with Michael has deeply influenced my research throughout my career, and I am grateful for that.

Current Distribution and Random Matrix Ensembles for an Integrable Asymmetric Fragmentation Process

“…Despite their greatly reduced complexity they capture various generic nonequilibrium phenomena such as the occurrence of shocks and particle condensation [3,4,5]. A major breakthrough in the exact calculation of universal properties came with the realization that various ensembles of random matrices occur in the study of current fluctuations and related quantities in the totally asymmetric simple exclusion process (TASEP) [6,7] and also in the polynuclear growth model [8,9,10,11].…”

A possible classification of nonequilibrium steady states

“…Even the simplest generalizations of thermal equilibrium, namely, non-equilibrium steady states (NESS), are currently analyzed case-by-case. Thus, much effort is directed at simple models, maintained far from equilibrium by imposing external driving forces, with the goal of identifying some generic classes of NESS [1,2]. Typically, these models are specified by a set of transition rates (motivated by physical considerations), so that the master equation provides a natural framework for analysis.…”

“…If the rates satisfy detailed balance, as for systems evolving towards thermal equilibrium where w j i /w i j = P eq i /P eq j , then all stationary currents vanish. An equivalent statement of detailed balance which does not reference P eq i explicitly [2] involves closed loops in configuration space, i.e.,…”

Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles

“…Rather it was found that there is exciting new physics in systems with more than one species of particles, including spontaneous symmetry breaking and phase separation phenomena, even in translation invariant systems [8,9] without the beneficial "assistance" of open boundaries or static defects [10] in facilitating phase transitions. It also emerged that similar phenomena may occur in single-species systems with one conservation law, provided there are kinetic constraints determined by a zero-rate condition somewhat analogous to the zero-temperature condition for long-range order in equilibrium systems with short-range interactions.…”