Theories and Applications of Mixed Quantum-Classical Non-adiabatic Dynamics

Electronically non-adiabatic processes are essential parts of photochemical process, collisions of excited species, electron transfer processes, and quantum information processing. Various non-adiabatic dynamics methods and their numerical implementation have been developed in the last decades. This review summarizes the most significant development of mixed quantum-classical methods and their applications which mainly include the Liouville equation, Ehrenfest mean-field, trajectory surface hopping, and multiple spawning methods. The recently developed quantum trajectory mean-field method that accounts for the decoherence corrections in a parameter-free fashion is discussed in more detail.     

Monte Carlo simulations of oscillations, chaos and pattern formation in heterogeneous catalytic reactions

Experimental studies employing surface science methods indicate that kinetic oscillations, chaos, and pattern formation in heterogeneous catalytic reactions often result from the interplay of rapid chemical reaction steps and relatively slow complementary processes such as oxide formation or adsorbate-induced surface restructuring. In general, the latter processes should be analysed in terms of theory of phase transitions. Therefore, the conventional mean-field reaction–diffusion equations widely used to describe oscillations in homogeneous reactions are strictly speaking not applicable. Under such circumstances, application of the Monte Carlo method becomes almost inevitable. In this review, we discuss the advantages and limitations of employing this technique and show what can be achieved in this way. Attention is focused on Monte Carlo simulations of CO oxidation on (1 0 0) and (1 1 0) single-crystal Pt and polycrystal Pt, Pd and Ir surfaces and of NO reduction by CO and H₂ on Pt(1 0 0). CO oxidation on supported nanometre-sized catalyst particles and NO reduction on composite catalysts are also discussed. The results show that with current computer facilities the MC technique has become an effective tool for analysing temporal oscillations and pattern formation on the nanometre scale in catalytic reactions occurring on both single crystals and supported particles.     

Modeling the social obesity epidemic with stochastic networks

In this paper we extend a compartmental model to the case of a homogenous network epidemic model for a study of the dynamics of obese populations. The social epidemic network-based approach developed here uses different algorithms and points of views regarding the simulation of the dynamics of the network. First, Monte Carlo simulations for homogeneous networks using a traditional constant probability transition rates and a mean-field-like approach are presented. We show that these networks evolve towards an approximately stationary state, which coincides with the one obtained by the underlying classical compartmental continuous model. A mean-field-like approach is applied in order to reduce the large computation time required when dealing with large contact networks. We also investigate, using homogenous contact networks, the effect of the realistic assumption that the waiting times between subpopulations follow a gamma distribution instead of the traditional exponential distribution. It is concluded that careful attention must be paid to the distributions assumed for the state periods.     

Functional central limit theorems for a large network in which customers join the shortest of several queues

We consider N single server infinite buffer queues with service rate . Customers arrive at rate N, choose L queues uniformly, and join the shortest. We study the processes for large N, where R^N_t(k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known, see Vvedenskaya et al. [15], Mitzenmacher [12] and Graham [5]. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data R^N₀ satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the R^N in equilibrium under the stability condition <, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits lim _N lim _t= lim _t lim _N by a compactness-uniqueness method. We deduce a posteriori the CLT for R^N₀ under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit.Mathematics Subject Classification (2000):Primary: 60K35; Secondary: 60K25, 60B12, 60F05, 37C75, 37A30     

The ambivalent effect of lattice structure on a spatial game

The evolution of cooperation is studied in lattice-structured populations, in which each individual who adopts one of the following strategies ‘always defect’ (ALLD), ‘tit-for-tat’ (TFT), and ‘always cooperate’ (ALLC) plays the repeated Prisoner’s Dilemma game with its neighbors according to an asynchronous update rule. Computer simulations are applied to analyse the dynamics depending on major parameters. Mathematical analyses based on invasion probability analysis, mean-field approximation, as well as pair approximation are also used. We find that the lattice structure promotes the evolution of cooperation compared with a non-spatial population, this is also confirmed by invasion probability analysis in one dimension. Meanwhile, it also inhibits the evolution of cooperation due to the advantage of being spiteful, which indicates the key role of specific life-history assumptions. Mean-field approximation fails to predict the outcome of computer simulations. Pair approximation is accurate in two dimensions but fails in one dimension.     

Mean-field limit for the stochastic Vicsek model

We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behaviour of individuals with a fixed speed. We rigorously derive the kinetic mean-field partial differential equation satisfied when the number N of particles tends to infinity, quantifying the convergence of the law of one particle to the solution of the PDE. For this we adapt a classical coupling argument to the present case in which both the particle system and the PDE are defined on a surface rather than on the whole space R^d. As part of the study we give existence and uniqueness results for both the particle system and the PDE.     

Non-linear Relaxation of the Photo-induced High Spin State in Spin-Crossover Solids: Effect of Correlations

We investigate a microscopic model based on the pair approximation, for the dynamical properties of the photo-induced high-spin state in spin-crossover solids at low temperature. The model uses the Ising-like hamiltonian and combines long- and short-range interactions. The stochastic treatment of the latter provides a set of two coupled differential equations for the macroscopic short- and long-range order parameters, which reproduce successfully the main features of the experimental relaxation curves.     

Symmetry-breaking vortex-lattice of a binary superfluid in a rotating bucket

We study spontaneous-symmetry-broken phase-separated vortex lattice in a weakly interacting uniform rapidly rotating binary Bose superfluid contained in a quasi-two-dimensional circular or square bucket. For the inter-species repulsion above a critical value, the two superfluid components separate and form a demixed phase with practically no overlap in the vortex lattices of the two components, which will permit an efficient experimental observation of such vortices and study their properties. In case of a circular bucket with equal intra-species energies of the two components, the two components separate into two non-overlapping semicircular domains for all frequencies of rotation Ω generating distinct demixed vortex lattices. In case of a binary Bose superfluid in both circular and square buckets, (a) the number of vortices increases linearly with Ω in agreement with a suggestion by Feynman, and (b) the rotational energy in the rotating frame decreases quadratically with Ω in agreement with a suggestion by Fetter.