Universal cumulants of the current in diffusive systems on a ring.

We calculate exactly the first cumulants of the integrated current and of the activity (which is the total number of changes of configurations) of the symmetric simple exclusion process on a ring with periodic boundary conditions. Our results indicate that for large system sizes the large deviation functions of the current and of the activity take a universal scaling form, with the same scaling function for both quantities. This scaling function can be understood either by an analysis of Bethe ansatz equations or in terms of a theory based on fluctuating hydrodynamics or on the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim.

Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization.

We consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves. For one particular model that we shall call the exponential model, the properties of the traveling wave front can be calculated exactly, as well as the statistics of the genealogy of the population. One striking result is that, for this particular model, the genealogical trees have the same statistics as the trees of replicas in the Parisi mean-field theory of spin glasses. We also find that in the exponential model, the coalescence times along these trees grow like the logarithm of the population size. A phenomenological picture of the propagation of wave fronts that we introduced in a previous work, as well as our numerical data, suggest that these statistics remain valid for a larger class of models, while the coalescence times grow like the cube of the logarithm of the population size.

Phenomenological theory giving the full statistics of the position of fluctuating pulled fronts.

We propose a phenomenological description for the effect of a weak noise on the position of a front described by the Fisher-Kolmogorov-Petrovsky-Piscounov equation or any other traveling-wave equation in the same class. Our scenario is based on four hypotheses on the relevant mechanism for the diffusion of the front. Our parameter-free analytical predictions for the velocity of the front, its diffusion constant and higher cumulants of its position agree with numerical simulations.

Distribution of current in nonequilibrium diffusive systems and phase transitions.

We consider diffusive lattice gases on a ring and analyze the stability of their density profiles conditionally to a current deviation. Depending on the current, one observes a phase transition between a regime where the density remains constant and another regime where the density becomes time dependent. Numerical data confirm this phase transition. This time dependent profile persists in the large drift limit and allows one to understand on physical grounds the results obtained earlier for the totally asymmetric exclusion process on a ring.

Current fluctuations in nonequilibrium diffusive systems: an additivity principle.

We formulate a simple additivity principle allowing one to calculate the whole distribution of current fluctuations through a large one dimensional system in contact with two reservoirs at unequal densities from the knowledge of its first two cumulants. This distribution (which in general is non-Gaussian) satisfies the Gallavotti-Cohen symmetry and generalizes the one predicted recently for the symmetric simple exclusion process. The additivity principle can be used to study more complex diffusive networks including loops.

Phase transition in the ABC model.

Recent studies have shown that one-dimensional driven systems can exhibit phase separation even if the dynamics is governed by local rules. The ABC model, which comprises three particle species that diffuse asymmetrically around a ring, shows anomalous coarsening into a phase separated steady state. In the limiting case in which the dynamics is symmetric and the parameter q describing the asymmetry tends to one, no phase separation occurs and the steady state of the system is disordered. In the present work, we consider the weak asymmetry regime q=exp(-beta/N), where N is the system size, and study how the disordered state is approached. In the case of equal densities, we find that the system exhibits a second-order phase transition at some nonzero beta(c). The value of beta(c)=2pi square root 3 and the optimal profiles can be obtained by writing the exact large deviation functional. For nonequal densities, we write down mean-field equations and analyze some of their predictions.

Exact free energy functional for a driven diffusive open stationary nonequilibrium system.

We obtain the exact probability exp[-LF([rho(x)])] of finding a macroscopic density profile rho(x) in the stationary nonequilibrium state of an open driven diffusive system, when the size of the system L-->infinity. F, which plays the role of a nonequilibrium free energy, has a very different structure from that found in the purely diffusive case. As there, F is nonlocal, but the shocks and dynamic phase transitions of the driven system are reflected in nonconvexity of F, in discontinuities in its second derivatives, and in non-Gaussian fluctuations in the steady state.

Free energy functional for nonequilibrium systems: an exactly solvable case.

We consider the steady state of an open system in which there is a flux of matter between two reservoirs at different chemical potentials. For a large system of size N, the probability of any macroscopic density profile rho(x) is exp[-NF([rho])]; F thus generalizes to nonequilibrium systems the notion of free energy density for equilibrium systems. Our exact expression for F is a nonlocal functional of rho, which yields the macroscopically long range correlations in the nonequilibrium steady state previously predicted by fluctuating hydrodynamics and observed experimentally.

Probability distribution of the free energy of a directed polymer in a random medium.

We calculate exactly the first cumulants of the free energy of a directed polymer in a random medium for the geometry of a cylinder. By using the fact that the nth moment of the partition function is given by the ground-state energy of a quantum problem of n interacting particles on a ring of length L, we write an integral equation allowing to expand these moments in powers of the strength of the disorder gamma or in powers of n. For n small and n approximately (Lgamma)(-1/2), the moments take a scaling form which allows us to describe all the fluctuations of order 1/L of the free energy per unit length of the directed polymer. The distribution of these fluctuations is the same as the one found recently in the asymmetric exclusion process, indicating that it is characteristic of all the systems described by the Kardar-Parisi-Zhang equation in 1+1 dimensions.

On the genealogy of a population of biparental individuals.

If one goes backward in time, the number of ancestors of an individual doubles at each generation. This exponential growth very quickly exceeds the population size, when this size is finite. As a consequence, the ancestors of a given individual cannot be all different and most remote ancestors are repeated many times in any genealogical tree. The statistical properties of these repetitions in genealogical trees of individuals for a panmictic closed population of constant size N can be calculated. We show that the distribution of the repetitions of ancestors reaches a stationary shape after a small number G(c) approximately log N of generations in the past, that only about 80% of the ancestral population belongs to the tree (due to coalescence of branches), and that two trees for individuals in the same population become identical after G(c)generations have elapsed. Our analysis is easy to extend to the case of exponentially growing population.

Genetic distance and species formation in evolving populations.

We compare the behavior of the genetic distance between individuals in evolving populations for three stochastic models. In the first model reproduction is asexual and the distribution of genetic distances reflects the genealogical tree of the population. This distribution fluctuates greatly in time, even for very large populations. In the second model reproduction is sexual with random mating allowed between any pair of individuals. In this case, the population becomes homogeneous and the genetic distance between pairs of individuals has small fluctuations which vanish in the limit of an infinitely large population. In the third model reproduction is still sexual but instead of random mating, mating only occurs between individuals which are genetically similar to each other. In that case, the population splits spontaneously into species which are in reproductive isolation from one another and one observes a steady state with a continual appearance and extinction of species in the population. We discuss this model in relation to the biological theory of speciation and isolating mechanisms. We also point out similarities between these three models of evolving populations and the theory of disordered systems in physics.