Mean-field Condition Research Articles

Thermodynamic stability and critical points in multicomponent mixtures with structured interactions.

Theoretical work has shed light on the phase behavior of idealized mixtures of many components with random interactions. However, typical mixtures interact through particular physical features, leading to a structured, nonrandom interaction matrix of lower rank. Here, we develop a theoretical framework for such mixtures and derive mean-field conditions for thermodynamic stability and critical behavior. Irrespective of the number of components and features, this framework allows for a generally lower-dimensional representation in the space of features and proposes a principled way to coarse-grain multicomponent mixtures as binary mixtures. Moreover, it suggests a way to systematically characterize different series of critical points and their codimensions in mean-field. Since every pairwise interaction matrix can be expressed in terms of features, our work is applicable to a broad class of mean-field models.

The symmetry-preserving mean field condition for electrostatic correlations in bulk.

Accurate simulations of a condensed system of ions or polar molecules are concerned with proper handling of the involved electrostatics. For such a Coulomb system at a charged planar interface, the Coulomb interaction averaged over the lateral directions with preserved symmetry serves as a necessary constraint in building any accurate handling that reconciles a simulated singlet charge density with the corresponding macroscopic charge/dielectric response. At present, this symmetry-preserving mean field (SPMF) condition represented in the reciprocal space is conjectured to be necessary for a simulated bulk system to reproduce correctly the charge structure factor of the macroscopic bulk as well. In this work, we further examine analytically the asymptotic behavior of the charge structure factor at small wavenumbers for an arbitrary charge-charge interaction. In light of our theoretical predictions, simulations with lengths of nearly 0.1μm are carried out to demonstrate that typical efficient methods violating the SPMF condition, indeed, fail to capture the exact charge correlations at small wavenumbers for both ionic and polar systems. However, for both types of systems, these existing methods can be simply amended to match the SPMF condition and subsequently to precisely probe the electrostatic correlations at all length scales.

Anisotropic oriented percolation in high dimensions

In this paper we study anisotropic oriented percolation on $\mathbb{Z}^d$ for $d\geq 4$ and show that the local condition for phase transition is closely related to the mean-field condition. More precisely, we show that if the sum of the local probabilities is strictly greater than one and each probability is not too large, then percolation occurs.

How spatial structure and neighbor uncertainty promote mutualists and weaken black queen effects

The ubiquity of cooperative cross-feeding (a resource-exchange mutualism) raises two related questions: Why is cross-feeding favored over self-sufficiency, and how are cross-feeders protected from non-producing cheaters? The Black Queen Hypothesis suggests that if leaky resources are costly, then there should be selection for either gene loss or self-sufficiency, but selection against mutualistic inter-dependency. Localized interactions have been shown to protect mutualists against cheaters, though their effects in the presence of self-sufficient organisms are not well understood. Here we develop a stochastic spatial model to examine how spatial effects alter the predictions of the Black Queen Hypothesis. Microbes need two essential resources to reproduce, which they can produce themselves (at a cost) or take up from neighbors. Additionally, microbes need empty sites to give birth into. Under well mixed mean-field conditions, the cross-feeders will always be displaced by a non-producer and a self-sufficient microbe. However, localized interactions have two effects that favor production. First, a microbe that interacts with a small number of neighbors will not always receive the essential resources it needs; this effect slightly harms cross-feeders but greatly harms non-producers. Second, microbes tend to displace other microbes that produce resources they need; this effect also slightly harms cross-feeders but greatly harms non-producers. Our work therefore suggests localized interactions produce an accelerating cost of non-production. Thus, the right trade-off between the cost of producing resources and the cost of sometimes being resource-limited can favor mutualistic inter-dependence over both self-sufficiency and non-production.

Decomposition of mean-field Gibbs distributions into product measures

We show that under a low complexity condition on the gradient of a Hamiltonian, Gibbs distributions on the Boolean hypercube are approximate mixtures of product measures whose probability vectors are critical points of an associated mean-field functional. This extends a previous work by the first author. As an application, we demonstrate how this framework helps characterize both Ising models satisfying a mean-field condition and the conditional distributions which arise in the emerging theory of nonlinear large deviations, both in the dense case and in the polynomially-sparse case.

Mean field conditions for coalescing random walks

The main results in this paper are about the full coalescence time $\mathsf{C}$ of a system of coalescing random walks over a finite graph $G$. Letting $\mathsf{m}(G)$ denote the mean meeting time of two such walkers, we give sufficient conditions under which $\mathbf{E}[\mathsf{C}]\approx 2\mathsf{m}(G)$ and $\mathsf{C}/\mathsf{m}(G)$ has approximately the same law as in the "mean field" setting of a large complete graph. One of our theorems is that mean field behavior occurs over all vertex-transitive graphs whose mixing times are much smaller than $\mathsf{m}(G)$; this nearly solves an open problem of Aldous and Fill and also generalizes results of Cox for discrete tori in $d\geq2$ dimensions. Other results apply to nonreversible walks and also generalize previous theorems of Durrett and Cooper et al. Slight extensions of these results apply to voter model consensus times, which are related to coalescing random walks via duality. Our main proof ideas are a strengthening of the usual approximation of hitting times by exponential random variables, which give results for nonstationary initial states; and a new general set of conditions under which we can prove that the hitting time of a union of sets behaves like a minimum of independent exponentials. In particular, this will show that the first meeting time among $k$ random walkers has mean $\approx\mathsf{m}(G)/\bigl({\matrix{k 2}}\bigr)$.

EFFECTIVE MEAN-FIELD THEORY BASED ON CUMULANT EXPANSION IN TREATING CLASSICAL HEISENBERG MODEL ON STACKED TRIANGULAR LATTICE

An effective mean-field theory based on cumulant expansion was used to deal with antiferromagnetic Heisenberg model on planar triangular lattice. The corrections of expansion were performed to the third order. By using the equation of the mean field condition, curves of internal energy E specific heat C staggered helicity K (order parameter) and the variational ratio of staggered helicity X were obtained when the proper values of effective external field were achieved. The calculated results showed that there were two phases (which were ordered antiferromagnetic phase and the disordered phase) in the spin system. The first order critical point is -kTc/Js = 1.65, the second is -kTc/Js = 1.35 and the third is -kTc/Js = 1.29, obviously closer to that of Monte Carlo simulation order by order. And also, the analytic expansion curves derived from this method exhibited higher proximity order by order to the Monte Carlo simulation. Such results showed that this method was a useful tool to obtain thermodynamically observables of spin system.

Mean-Field Conditions for Percolation on Finite Graphs

Let {G n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G n | = n. Denote by p t(v, v) the return probability after t steps of the non-backtracking random walk on G n . We show that if p t(v, v) has quasi-random properties, then critical bond-percolation on G n behaves as it would on a random graph. More precisely, if $$\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$$then the size of the largest component in p-bond-percolation with \({p =\frac{1+O(n^{-1/3})}{d-1}}\) is roughly n 2/3. In Physics jargon, this condition implies that there exists a scaling window with a mean-field width of n −1/3 around the critical probability \({p_c =\frac{1}{d-1}}\).A consequence of our theorems is that if {G n } is a transitive expander family with girth at least \({\left( {\frac{2}{3} + \epsilon} \right) {\rm log}_{d - 1}n}\) then {G n } has the above scaling window around \({p_c =\frac{1}{d-1}}\). In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak [LuPS] has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation.

Athermal mechanisms of size-dependent crystal flow gleaned from three-dimensional discrete dislocation simulations

Recent experimental studies have revealed that micrometer-scale face-centered cubic (fcc) crystals show strong strengthening effects, even at high initial dislocation densities. We use large-scale three-dimensional discrete dislocation simulations (DDS) to explicitly model the deformation behavior of fcc Ni microcrystals in the size range of 0.5–20 μm. This study shows that two size-sensitive athermal hardening processes, beyond forest hardening, are sufficient to develop the dimensional scaling of the flow stress, stochastic stress variation, flow intermittency and high initial strain-hardening rates, similar to experimental observations for various materials. One mechanism, source-truncation hardening, is especially potent in micrometer-scale volumes. A second mechanism, termed exhaustion hardening, results from a breakdown of the mean-field conditions for forest hardening in small volumes, thus biasing the statistics of ordinary dislocation processes.

EFFECTS OF ATOMISTIC DOMAIN SIZE ON HYBRID LATTICE BOLTZMANN–MOLECULAR DYNAMICS SIMULATIONS OF DENSE FLUIDS

We present a convergence study for a hybrid Lattice Boltzmann-Molecular Dynamics model for the simulation of dense liquids. Time and length scales are decoupled by using an iterative Schwarz domain decomposition algorithm. The velocity field from the atomistic domain is introduced as forcing terms to the Lattice Boltzmann model of the continuum while the mean field of the continuum imposes mean field conditions for the atomistic domain. In the present paper we investigate the effect of varying the size of the atomistic subdomain in simulations of two dimensional flows of liquid argon past carbon nanotubes and assess the efficiency of the method.

Stability and correlations in dilute two-dimensional boson systems

The hyperspherical adiabatic expansion method is used to describe correlations in a symmetric boson system rigorously confined to two spatial dimensions. The hyperangular eigenvalue equation turns out to be almost independent of the hyperradius, whereas the solutions are strongly varying with the strength of the attractive two-body potentials. Instability is encountered in hyperangular, hyperradial, and mean-field equations for almost identical strengths inversely proportional to the particle number. The derived conditions for stability are similar to mean-field conditions and closely related to the possible occurrence of the Thomas and Efimov effects. Renormalization in mean-field calculations for two spatial dimensions is probably not needed.

On optimal mean-field descriptions in finite-temperature many-body theories: Use of thermal Brillouin and Bruckner conditions

Although the structural similarity between the properties of a thermal trace and the zero-temperature expectation values for quantum systems has been known for quite some time, not all the practical computational methods in the thermal field theories exploit this correspondence explicitly. Using a thermal field theory derived by us, which introduces the thermal analogues of normal ordering and Wick’s expansion, a very close resemblance between zero-temperature and finite-temperature field theories can be established. We use this apparatus in this paper to derive optimal conditions of mean-field and correlated descriptions of thermally averaged quantities. It is shown that the optimal mean-field conditions for the free energy is equivalent to the minimum value for the average energy. The optimality conditions turn out to be exact thermal analogues of the Brillouin conditions. The optimal mean-field condition to generate the minimum value of the ratio Z/Zo, where Z and Zo are the exact and the mean-field partition functions, yields the exact thermal analogue of the Bruckner condition. In a similar vein, we generalize the thermal Brillouin condition to include correlated functions used in the evaluation of the thermal trace. The optimal choice of the correlated ground states leads to many-particle generalizations of the thermal Brillouin conditions. In the context of the path-integral methods for determining Z, we envisage use of a local optimal mean field that depends on each point on the path — leading to local thermal analogues of Brillouin and Bruckner conditions. To derive these conditions, we have used another apparatus of field theory derived recently by us that uses concepts of normal ordering and Wick expansion with respect to a path-integral measure (rather than the measure implied by thermal trace). We hope to demonstrate in our discussions that this way of formulating thermal many-body problems enables us to exploit deep similarities between thermal and zero-temperature situations which are difficult to discern in the traditional methods. Illustrative examples by way of deriving thermal Brillouin and Bruckner conditions for typical Fermionic and Bosonic problems are presented.

Collective Behaviors in Spatially Extended Systems with Local Interactions and Synchronous Updating

Assessing the extent to which dynamical systems with many degrees of freedom can be described within a thermodynamics formalism is a problem that currently attracts much attention. In this context, synchronously updated regular lattices of identical, chaotic elements with local interactions are promising models for which statistical mechanics may be hoped to provide some insights. This article presents a large class of cellular automata rules and coupled map lattices of the above type in space dimensions d=2 to 6. Such simple models can be approached by a mean-field approximation which usually reduces the dynamics to that of a map governing the evolution of some extensive density. While this approxima­ tion is exact in the d== limit, where macroscopic variables must display the time· dependent behavior of the mean·field map, basic intuition from equilibrium statistical mechanics rules out any such behavior in a low·dimensional system, since it would involve the collective motion of locally disordered elements. The models studied are chosen to be as close as possible to mean-field conditions, i.e., rather high space dimension, large connectivity, and equal·weight coupling between sites. While the mean·field evolution is never observed, a new type of non·trivial collective behavior is found, at odds with the predictions of equilibrium statistical mechanics. Both in the cellular automata models and in the coupled map lattices, macroscopic variables frequently display a non·transient, time· dependent, low·dimensional dynamics emerging out of local disorder. Striking examples are period 3 cycles in two-state cellular automata and a Hopf bifurcation for a d=5 lattice of coupled logistic maps. An extensive account of the phenomenology is given, including a catalog of behaviors, classification tables for the cellular automata rules, and bifurcation diagrams for the coupled map lattices. The observed underlying' dynamics is accompanied by an intrinsic quasi·Gaussian noise (stem, ming from the local disorder) which disappears in the infinite-size limit. The collective behaviors constitute a robust phenomenon, resisting external noise, small changes in the local dynamics, and modifi,cations of the initial and boundary conditions. Synchronous updating, high space dimension and the regularity of connections are shown to be crucial ingredients in the subtle build-up of correlations giving rise to the collective motion. The discussion stresses the need for a theoretical understanding that neither equilibrium statistical mechanics nor higher-order mean·field approxima· tions are able to provide.

Modelling the onset of life

We consider various factors that influence the growth of DNA-like strands from a “soup” of monomers and dimers. The growth of polymers in a cell-like environment is compared to the growth of a typical chain in ideal or mean-field conditions to determine the effects of the environment on diversity. The results obtained are relatively insensitive to the fine details of the model. A feasible way is proposed in which chains of random monomers could evolve to encode information in sequences along their length.

On the stability of a new relativistic kinetic equation

Recently, Israel and Kandrup have formulated a new relativistic kinetic equation appropriate for a self-gravitating system. That equation allos for the combined influences of (i) collective mean fields, which define an ''average'' spacetime geometry, and (ii) ''fluctuations'' from mean field conditions, which are modeled in a Landau-type approximation. In the limit that the fluctuations are compleely ignored, one recovers the mean field theory discussed by such authors as Ipser and Thorne. It is shown here that this n ew kinetic equation admits the relativistic isothermal distribution as an exact stationary solution. Linearized perturbation equations are then derived for small spherically symmetric departures from the isothermal ''equilibrium.'' Attention next focuses upon an ''obit-averaged'' version of the kinetic equation which eliminates from explicit consideration the collective mean field effects. One finds, in analogy with the Newtonian theory considered by Ipser and Kandrup, that the isothermal solution to this averaged equation will be stable with respect to linearized spherically symmetric perturbtions if and only if the mean field Boltzmann entropy, which is extremized by the isothermal distribution, is in fact a local maximum. Finally, this result is shown to be a simple manifestation of a more general H-theorem inequality.