The Mpemba effect is a counter-intuitive relaxation phenomenon, where a system prepared at a hot temperature cools down faster than an identical system initiated at a cold temperature when both are quenched to an even colder bath. Such non-monotonic relaxations were observed in various systems, including water, magnetic alloys, polymers and driven granular gases. We analyze the Mpemba effect in Markovian dynamics and discover that a stronger version of the effect often exists for a carefully chosen set of initial temperatures. In this strong Mpemba effect, the relaxation time jumps to a smaller value leading to exponentially faster equilibration dynamics. The number of such special initial temperatures defines the Mpemba index, whose parity is a topological property of the system. To demonstrate these concepts, we first analyze the different types of Mpemba relaxations in the mean field anti-ferromagnet Ising model, which demonstrates a surprisingly rich Mpemba phase diagram. Moreover, we show that the strong effect survives the thermodynamic limit, and that it is tightly connected with thermal overshoot -in the relaxation process, the temperature of the relaxing system can decay non-monotonically as a function of time. Using the parity of the Mpemba index, we then study the occurrence of the strong Mpemba effect in a large class of thermal quench processes and show that it happens with non-zero probability even in the thermodynamic limit. This is done by introducing the isotropic model for which we obtain analytical lower bound estimates for the probability of the strong Mpemba effects. Consequently, we expect that such exponentially faster relaxations can be observed experimentally in a wide variety of systems.

The late-time distribution function P (x,t) of a particle diffusing in a one-dimensional logarithmic potential is calculated for arbitrary initial conditions. We find a scaling solution with three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ∼ t 1/2 ) and a subdiffusive (x ∼ t γ with a given γ < 1/2) length scale, respectively, (ii) the overall scaling function is selected by the initial condition, and (iii) depending on the tail of the initial condition, the scaling exponent that characterizes the scaling function is found to exhibit a transition from a continuously varying to a fixed value.

The impact of temporally correlated dynamics on nonequilibrium condensation is studied using a non-Markovian zero-range process (ZRP). We find that memory effects can modify the condensation scenario significantly: (i) For mean-field dynamics, the steady state corresponds to that of a Markovian ZRP, but with modified hopping rates which can affect condensation; (ii) for nearest-neighbor hopping dynamics in one dimension, the condensate is found to occupy two adjacent lattice sites and to drift with a finite velocity. The validity of these results in a more general context is discussed.

Abstract. We study the asymmetric zero-range process (ZRP) with L sites and open boundaries, conditioned to carry an atypical current. Using a generalized Doob h-transform we compute explicitly the transition rates of an effective process for which the conditioned dynamics are typical. This effective process is a zero-range process with renormalized hopping rates, which are space dependent even when the original rates are constant. This leads to non-trivial density profiles in the steady state of the conditioned dynamics, and, under generic conditions on the jump rates of the unconditioned ZRP, to an intriguing supercritical bulk region where condensates can grow. These results provide a microscopic perspective on macroscopic fluctuation theory (MFT) for the weakly asymmetric case: It turns out that the predictions of MFT remain valid in the non-rigorous limit of finite asymmetry. In addition, the microscopic results yield the correct scaling factor for the asymmetry that MFT cannot predict.

Abstract. Condensation transition in a non-Markovian zero-range process is studied in one and higher dimensions. In the mean-field approximation, corresponding to infinite range hopping, the model exhibits condensation with a stationary condensate, as in the Markovian case, but with a modified phase diagram. In the case of nearestneighbor hopping, the condensate is found to drift by a "slinky" motion from one site to the next. The mechanism of the drift is explored numerically in detail. A modified model with nearest-neighbor hopping which allows exact calculation of the steady state is introduced. The steady state of this model is found to be a product measure, and the condensate is stationary.

Abstract. The equation which describes a particle diffusing in a logarithmic potential arises in diverse physical problems such as momentum diffusion of atoms in optical traps, condensation processes, and denaturation of DNA molecules. A detailed study of the approach of such systems to equilibrium via a scaling analysis is carried out, revealing three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ∼ √ t) and a subdiffusive (x ≪ √ t) length scales, respectively; (ii) the scaling exponents and scaling functions corresponding to both regimes are selected by the initial condition; and (iii) this dependence on the initial condition manifests a "phase transition" from a regime in which the scaling solution depends on the initial condition to a regime in which it is independent of it. The selection mechanism which is found has many similarities to the marginal stability mechanism which has been widely studied in the context of fronts propagating into unstable states. The general scaling forms are presented and their practical and theoretical applications are discussed.

We examine the effect of spatial correlations on the phenomenon of real-space condensation in driven masstransport systems. We suggest that in a broad class of models with a spatially correlated steady state, the condensate drifts with a nonvanishing velocity. We present a robust mechanism leading to this condensate drift. This is done within the framework of a generalized zero-range process (ZRP) in which, unlike the usual ZRP, the steady state is not a product measure. The validity of the mechanism in other mass-transport models is discussed.

* O. Hirschberg and S. Reuveni had equal contribution to this workThe Asymmetric Simple Inclusion Process (ASIP), a lattice-gas model of unidirectional transport and aggregation, was recently proposed as an 'inclusion' counterpart of the Asymmetric Simple Exclusion Process (ASEP). In this paper we present an exact closed-form expression for the probability that a given number of particles occupies a given set of consecutive lattice sites. Our results are expressed in terms of the entries of Catalan's trapezoids -number arrays which generalize Catalan's numbers and Catalan's triangle. We further prove that the ASIP is asymptotically governed by: (i) an inverse square root law of occupation; (ii) a square root law of fluctuation; and (iii) a Rayleigh law for the distribution of inter-exit times. The universality of these results is discussed.