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Abstract:: A new model of evolution is presented for finite size systems. Conditions under which a minority species can emerge, spread and stabilize to a macroscopic size are studied. It is found that space organization is instrumental in addition to a qualitative advantage. Some peculiar topologies ensure the overcome of the initial majority species. However the probability of such local clusters is very small and depend strongly on the system size. A probabilistic phase diagram is obtained for small sizes. It reduces to a trivial situation in the thermodynamic limit, thus indicating the importance of dealing with finite systems in evolution problems. Results are discussed with respect to both Darwin and punctuated equilibria theories.

In many equilibrium or nonequilibrium statistical physics problems, fluctuations play a crucial role. Often those problems are too complex to be solved analytically. Accordingly, numerical algorithms keeping track of the fluctuations are needed. Cellular automata (CA) and Lattice Boltzmann (LB) models are two possible approaches to simulate complex systems. CA models keep track of many-body correlations and provide a description of the fluctuations. However, they lead to a noisy dynamics and impose a restriction on the possible values of the viscosity. On the other hand, LB models are numerically more efficient and offer much more flexibility to adjust the physical parameters, but they neglect the fluctuations. We have developed a new multiparticle lattice model which reconciles both approaches. The main characteristics of this approach are explained, and our model is used to study the kinetics of two-dimensional ballistic annihilation.

The effect of dissociation of the invading electrolyte on the formation of Liesegang bands is investigated. We find, using organic compounds with known dissociation constants, that the spacing coefficient, 1 + p, that characterizes the position of the nth band as xn ~ (1 + p)ⁿ decreases with increasing dissociation constant, Kd. Theoretical arguments are developed to explain these experimental findings and to calculate explicitly the Kd dependence of 1 + p.

We consider a new cellular automata rule for a synchronous random walk on a two-dimensional square lattice, subject to an exclusion principle. It is found that the macroscopic behavior of our model obeys the telegraphists's equation, with an adjustable diffusion constant. By construction, the dynamics of our model is exactly described by a linear discrete Boltzmann equation which is solved analytically for some boundary conditions. Consequently, the connection between the microscopic and the macroscopic descriptions is obtained exactly and the continuous limit studied rigorously. The typical system size for which a true diffusive behavior is observed may be deduced as a function of the parameters entering into the rule. It is shown that a suitable choice of these parameters allows us to consider quite small systems. In particular, our cellular automata model can simulate the Laplace equation to a precision of the order (?/L)6, whereL is the size of the system and? the lattice spacing. Implementation of this algorithm on special-purpose machines leads to the fastest way to simulate diffusion on a lattice.

The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schrödinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and q-deformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions, and finite-size scaling are also discussed in detail.Copyright 1994, 1999 Academic Press, Inc

The structure factor S(q) of a simple model describing adsorption on a two-dimensional surface is computed by real-space renormalization-group technique for T⩾ Tc. The method used is based on first-order cumulant expansion implemented with an ad hoc recursion relation for the coupling constants. The model is equivalent to an antiferromagnetic Ising model in an external field.

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Abstract:: A simple cellular automata model for a two-group war over the same “territory” is presented. It is shown that a qualitative advantage is not enough for a minority to win. A spatial organization as well a definite degree of aggressiveness are instrumental to overcome a less fitted majority. The model applies to a large spectrum of competing groups: smoker-non smoker war, epidemic spreading, opinion formation, competition for industrial standards and species evolution. In the last case, it provides a new explanation for punctuated equilibria.