Phase Transitions In Statistical Mechanics Research Articles

Emergent Criticality in Coupled Boolean Networks

Early embryonic development involves forming all specialized cells from a fluid-like mass of identical stem cells. The differentiation process consists of a series of symmetry-breaking events, starting from a high-symmetry state (stem cells) to a low-symmetry state (specialized cells). This scenario closely resembles phase transitions in statistical mechanics. To theoretically study this hypothesis, we model embryonic stem cell (ESC) populations through a coupled Boolean network (BN) model. The interaction is applied using a multilayer Ising model that considers paracrine and autocrine signaling, along with external interventions. It is demonstrated that cell-to-cell variability can be interpreted as a mixture of steady-state probability distributions. Simulations have revealed that such models can undergo a series of first- and second-order phase transitions as a function of the system parameters that describe gene expression noise and interaction strengths. These phase transitions result in spontaneous symmetry-breaking events that generate new types of cells characterized by various steady-state distributions. Coupled BNs have also been shown to self-organize in states that allow spontaneous cell differentiation.

Statistical mechanics of phase transitions in elastic media with vanishing thermal expansion.

We consider a minimal spin model for Ising transitions in an isotropic elastic medium in the zero thermal expansion (ZTE) limit. We set up the elastic theory for this system. We use this theory to identify and study the nature of the fluctuations in the system near the second order phase transitions at T_{c} in the ZTE limit given by dT_{c}/dV=0, where V is the system volume, and explore anomalous elasticity. Allowing for the local strain to couple asymmetrically or selectively with the states of the order parameter, we uncover the dramatic effects of these couplings on the fluctuations of the local displacements near T_{c}, and also on the nature of the phase transition itself. Near second-order phase transitions and with weak asymmetry in the order parameter-strain couplings, the variance of the displacement fluctuations in two dimensions scale with the system size L in a universal fashion as [ln(L/a_{0})]^{2/3}; a_{0} is a small-scale cutoff. Likewise, the correlation functions of the difference of the local displacements at two different points separated by r scale as [ln(r/a_{0})]^{2/3} for large r. For stronger selectivity above a finite threshold, this variance diverge as L exceeds beyond a (nonuniversal) size, determined by the model parameters, signaling a transition to a phase with only short-range order or the loss of the positional order of the elastic medium. At dimensions higher than two, for sufficiently weak selectivity, the variance of the displacement fluctuations is L-independent corresponding to long-range order. However, if the selectivity parameters rise beyond a dimension-dependent threshold value, then again the positional order is lost with a concomitant transition to a phase with short-range order. Large values of the order parameter-strain couplings can turn the phase transition into a first order as well. Our theory establishes a one-to-one correspondence between the order of phase transitions and anomalous elasticity near the transitions. Our theory should be a useful guide to possible synthesis of appropriate ZTE materials.

Phase transitions and phase coexistence: equilibrium systems versus externally driven or active systems - Some perspectives

ABSTRACT A tutorial introduction to the statistical mechanics of phase transitions and phase coexistence is presented, starting out from equilibrium systems and nonequilibrium steady-state situations in externally driven systems, such as unmixing of sheared binary fluid mixtures, the driven lattice gas model, and the onset of Rayleigh-Bénard convection. Then, some models for phase separation in models for active systems, where particles possess internal motility, are discussed, emphasizing what one can learn by extending analysis methods to study phase transitions in equilibrium systems by computer simulations to active systems. Specific examples will include colloid-polymer mixtures where the colloids are assumed to be active particles, and active Brownian particles. The extent to which concepts familiar from the study of equilibrium systems are still useful will be critically discussed.

On the dimension of the space of harmonic functions on transitive shift spaces

In this paper, we show a new relation between phase transition in Statistical Mechanics and the dimension of the space of harmonic functions (SHF) for a transfer operator. This is accomplished by extending the classical Ruelle-Perron-Frobenius theory to the realm of low regular potentials defined on either finite or infinite (uncountable) alphabets. We also give an example of a potential having a phase transition where the Perron-Frobenius eigenvector space has dimension two. We discuss entropy and equilibrium states, in this general setting, and show that if the SHF is non-trivial, then the associated equilibrium states have full support. We also obtain a weak invariance principle in cases where the spectral gap property is absent. As a consequence, a functional central limit theorem for non-local observables of the Dyson model is obtained.

Scaling Behaviour and Critical Phase Transitions in Integrated Information Theory

Integrated Information Theory proposes a measure of conscious activity (), characterised as the irreducibility of a dynamical system to the sum of its components. Due to its computational cost, current versions of the theory (IIT 3.0) are difficult to apply to systems larger than a dozen units, and, in general, it is not well known how integrated information scales as systems grow larger in size. In this article, we propose to study the scaling behaviour of integrated information in a simple model of a critical phase transition: an infinite-range kinetic Ising model. In this model, we assume a homogeneous distribution of couplings to simplify the computation of integrated information. This simplified model allows us to critically review some of the design assumptions behind the measure and connect its properties with well-known phenomena in phase transitions in statistical mechanics. As a result, we point to some aspects of the mathematical definitions of IIT that 3.0 fail to capture critical phase transitions and propose a reformulation of the assumptions made by integrated information measures.

Ising model for melt ponds on Arctic sea ice

Perhaps the most iconic feature of melting Arctic sea ice is the distinctive ponds that form on its surface. The geometrical patterns describing how melt water is distributed over the surface largely determine the solar reflectance and transmittance of the sea ice cover, which are key parameters in climate modeling and upper ocean ecology. In order to help develop a predictive theoretical approach to studying melting sea ice, and the resulting patterns of light and dark regions on its surface in particular, we look to the statistical mechanics of phase transitions and introduce a two-dimensional random field Ising model which accounts for only the most basic physics in the system. The ponds are identified as metastable states in the model, where the binary spin variable corresponds to the presence of melt water or ice on the sea ice surface. With the lattice spacing determined by snow topography data as the only measured parameter input into the model, energy minimization drives the system toward realistic pond configurations from an initially random state. The model captures the essential mechanism of pattern formation of Arctic melt ponds, with predictions that agree very closely with observed scaling of pond sizes and transition in pond fractal dimension.

Speculative and hedging interaction model in oil and U.S. dollar markets: Nash equilibria for one or more banks

We determine the (refined) Nash equilibria for a bounded rational Carfi'-Musolino speculative and hedging model. This model shows two types of operators: a real economic subject (Air) and one or more investment banks (Bank). We consider the bank agents' behavior to equilibrate much more quickly than that of Air, as they react to the move of Air. In this sense, Air is an acting external agent, whereas the action of the banks is `annealed' - i.e. , equilibrates before Air makes its next transaction. When Air makes no purchases of oil futures as a hedge, two Nash equilibria exist for the bank agents. However, a unique Nash equilibrium exists for the bank agents when Air makes a purchase. This is a result of Air's purchase breaking a symmetry of the potential. The existence of multiple equilibriums in this two-market model is in the spirit of the Sonnenschein-Mantel-Debreu theorem, and is associated with phase transitions in statistical mechanics.

Continuous and discontinuous phase transitions in the evolution of a polygenic trait under stabilizing selective pressure

The presence of phenomena analogous to phase transition in Statistical Mechanics has been suggested in the evolution of a polygenic trait under stabilizing selection, mutation and genetic drift. By using numerical simulations of a model system, we analyze the evolution of a population of N diploid hermaphrodites in random mating regime. The population evolves under the effect of drift, selective pressure in form of viability on an additive polygenic trait, and mutation. The analysis allows to determine a phase diagram in the plane of mutation rate and strength of selection. The involved pattern of phase transitions is characterized by a line of critical points for weak selective pressure (smaller than a threshold), whereas discontinuous phase transitions, characterized by metastable hysteresis, are observed for strong selective pressure. A finite-size scaling analysis suggests the analogy between our system and the mean-field Ising model for selective pressure approaching the threshold from weaker values. In this framework, the mutation rate, which allows the system to explore the accessible microscopic states, is the parameter controlling the transition from large heterozygosity (disordered phase) to small heterozygosity (ordered one).

Large deviations, condensation and giant response in a statistical system

We study the probability distribution P of the sum of a large number of non-identically distributed random variables nm. Condensation of fluctuations, the phenomenon whereby one of such variables provides a macroscopic contribution to the global probability, is discussed and interpreted by analogy with phase-transitions in statistical mechanics. A general expression for P is derived, and its sensitivity to the details of the distribution of a single nm is worked out. These general results are verified by the analytical and numerical solution of some specific examples.

Emergence of Heavy-Tailed Distributions in a Random Multiplicative Model Driven by a Gaussian Stochastic Process

We consider a random multiplicative stochastic process with multipliers given by the exponential of a Brownian motion. The positive integer moments of the distribution function can be computed exactly, and can be represented as the grand partition function of an equivalent lattice gas with attractive 2-body interactions. The numerical results for the positive integer moments display a sharp transition at a critical value of the model parameters, which corresponds to a phase transition in the equivalent lattice gas model. The shape of the terminal distribution changes suddenly at the critical point to a heavy-tailed distribution. The transition can be related to the position of the complex zeros of the grand partition function of the lattice gas, in analogy with the Lee, Yang picture of phase transitions in statistical mechanics. We study the properties of the equivalent lattice gas in the thermodynamical limit, which corresponds to the continuous time limit of the random multiplicative model, and derive the asymptotics of the approach to the continuous time limit. The results can be generalized to a wider class of random multiplicative processes, driven by the exponential of a Gaussian stochastic process.

The Ising model and critical behavior of transport in binary composite media

We present a general theory for critical behavior of transport in binary composite media. The theory holds for lattice and continuum percolation models in both the static case with real parameters and the quasi–static case (frequency dependent) with complex parameters. Through a direct, analytic correspondence between the magnetization of the Ising model and the effective parameter problem of two phase random media, we show that the critical exponents of the transport coefficients satisfy the standard scaling relations for phase transitions in statistical mechanics. Our work also shows that delta components form in the underlying spectral measures at the spectral endpoints precisely at the percolation threshold pc and at 1 − pc. This is analogous to the Lee-Yang-Ruelle characterization of the Ising model phase transition, and identifies these transport transitions with the collapse of spectral gaps in these measures.

Threshold Effects in Parameter Estimation as Phase Transitions in Statistical Mechanics

Threshold effects in the estimation of parameters of nonlinearly modulated, continuous-time, wide-band waveforms, are examined from a statistical physics perspective. These threshold effects are shown to be analogous to phase transitions of certain disordered physical systems in thermal equilibrium. The main message, in this work, is in demonstrating that this physical point of view may be insightful for understanding the interactions between two or more parameters to be estimated, from the aspects of the threshold effect.

Phase behavior of polydisperse spheres: Simulation strategies and an application to the freezing transition

The statistical mechanics of phase transitions in dense systems of polydisperse particles presents distinctive challenges to computer simulation and analytical theory alike. The core difficulty, namely, dealing correctly with particle size fractionation between coexisting phases, is set out in the context of a critique of previous simulation work on such systems. Specialized Monte Carlo simulation techniques and moment free energy method calculations, capable of treating fractionation exactly, are then described and deployed to study the fluid-solid transition of an assembly of repulsive spherical particles described by a top-hat "parent" distribution of particle sizes. The cloud curve delineating the solid-fluid coexistence region is mapped as a function of the degree of polydispersity δ, and the properties of the incipient "shadow" phases are presented. The coexistence region is found to shift to higher densities as δ increases, but does not exhibit the sharp narrowing predicted by many theories and some simulations.

Metamodelling Messages Conveyed in Five Statistical Mechanical Textbooks from 1936 to 2001

Modelling is a significant aspect of doing physics and it is important how this activity is taught. This paper focuses on the explicit or implicit messages about modelling conveyed to the student in the treatments of phase transitions in statistical mechanics textbooks at beginning graduate level. Five textbooks from the 1930s to the present are analysed with respect to their messages about the following issues: What is a good model? What is the purpose of modelling? What does it mean to understand a natural phenomenon? It is argued that these texts give the student quite different perceptions of these issues and thus what of it means to do physics.

Water as an Active Constituent in Cell Biology

When Szent-Gyorgyi called water the “matrix of life”,1 he was echoing an old sentiment. Paracelsus in the 16th century said that “water was the matrix of the world and of all its creatures.”2 But Paracelsus’s notion of a matrixsan active substance imbued with fecund, life-giving propertiess was quite different from the picture that, until very recently, molecular biologists have tended to hold of water’s role in the chemistry of life. Although acknowledging that liquid water has some unusual and important physical and chemical propertiessits potency as a solvent, its ability to form hydrogen bonds, its amphoteric naturesbiologists have regarded it essentially as the backdrop on which life’s molecular components are arrayed. It used to be common practice, for example, to perform computer simulations of biomolecules in a vacuum. Partly this was because the computational intensity of simulating a polypeptide chain was challenging even without accounting for solvent molecules too, but it also reflected the prevailing notion that water does little more than temper or moderate the basic physicochemical interactions responsible for molecular biology. What Gerstein and Levitt said 9 years ago remains true today: “When scientists publish models of biological molecules in journals, they usually draw their models in bright colors and place them against a plain, black background”.3 Curiously, this neglect of water as an active component of the cell went hand in hand with the assumption that life could not exist without it. That was basically an empirical conclusion derived from our experience of life on Earth: environments without liquid water cannot sustain life, and special strategies are needed to cope with situations in which, because of extremes of either heat or cold, the liquid is scarce.4-6 The recent confirmation that there is at least one world rich in organic molecules on which rivers and perhaps shallow seas or bogs are filled with nonaqueous fluidsthe liquid hydrocarbons of Titan7smight now bring some focus, even urgency, to the question of whether water is indeed a * E-mail: p.ball@nature.com. Philip Ball is a science writer and a consultant editor for Nature, where he worked as an editor for physical sciences for more than 10 years. He holds a Ph.D. in physics from the University of Bristol, where he worked on the statistical mechanics of phase transitions in the liquid state. His book H2O: A Biography of Water (Weidenfeld & Nicolson, 1999) was a survey of the current state of knowledge about the behavior of water in situations ranging from planetary geomorphology to cell biology. He frequently writes about aspects of water science for both the popular and the technical media.

Two-dimensional chiral segregation of the bent hard needles model with Lennard-Jones sites

Assuming the aggregates of a single chiral component as a pure phase, chiral segregation can be considered as a coexistence of two phases, therefore the formalisms of phase transitions in Statistical Mechanics can be applied. That is, chiral segregation can be considered as phase equilibrium. The standard mechanism to understand phase equilibrium considers the complete van der Waals forces, including repulsions and attractions. However, during the last few years, a particular kind of fluid–fluid equilibrium, explained in terms of only excluded volume effects, has gained important interest. This kind of phenomenon is known as depletion forces. In this paper, the contribution of these two mechanisms is investigated. In order to analyze the effect that repulsions and attractions have on chiral segregation we study a two-dimensional model, bent hard needles model. In one case only infinitely repulsive interactions are considered while in the second case, one and two Lennard-Jones (LJ) sites are added to model the complete van der Waals forces. In the first case, because of the nature of the model, the cause for chiral segregation is due to excluded volume effects that appear when the density of the system is sufficiently high. In the second case, the model proposed is more complex and besides the density the temperature also affects the phase separation. Monte Carlo simulations in the Gibbs ensemble (GMC) are used to perform this comparison. As a result, the segregation phase diagram is obtained for the infinitely repulsive model and that with two LJ sites. The important conclusions are that for some particular molecular geometries, repulsions are not able to produce chiral segregation at all, independently of density. However, when attractions are included the effect of repulsion is complemented and phase separation can be achieved even for those molecular geometries where infinitely repulsions were not able.

Problemy fazovykh perekhodov v statisticheskoi mekhanike

Problemy fazovykh perekhodov v statisticheskoi mekhanike

Physical Chemistry of Equilibrium Polymerization

Reversible polymerizations occur in inorganic, organic, and biological molecules. The thermodynamic states at which monomer-to-polymer reactions become favorable can be viewed as second-order phase transitions and can be profitably treated by the statistical mechanics of phase transitions. The nature and development of the microscopic structure of a solution of active polymers in equilibrium with the monomers pose interesting issues for the physical chemist.

Critical phenomena in gravitational collapse

Abstract In general relativity black holes can be formed from regular initial data that do not contain a black hole already. The space of regular initial data for general relativity therefore splits naturally into two halves: data that form a black hole in the evolution and data that do not. The spacetimes that are evolved from initial data near the black hole threshold have many properties that are mathematically analogous to a critical phase transition in statistical mechanics. Solutions near the black hole threshold go through an intermediate attractor, called the critical solution. The critical solution is either time-independent (static) or scale-independent (self-similar). In the latter case, the final black hole mass scales as (p−p ∗ ) γ along any 1-parameter family of data with a regular parameter p such that p=p ∗ is the black hole threshold in that family. The critical solution and the critical exponent γ are universal near the black hole threshold for a given type of matter. We show how the essence of these phenomena can be understood using dynamical systems theory and dimensional analysis. We then review separately the analogy with critical phase transitions in statistical mechanics, and aspects specific to general relativity, such as spacetime singularities. We examine the evidence that critical phenomena in gravitational collapse are generic, and give an overview of their rich phenomenology.

Critical Behavior of Transport in Lattice and Continuum Percolation Models

It has been observed that the critical exponents of transport in the continuum, such as in the Swiss cheese and random checkerboard models, can exhibit nonuniversal behavior, with values different than the lattice case. Nevertheless, it is shown here that the transport exponents for both lattice and continuum percolation models satisfy the standard scaling relations for phase transitions in statistical mechanics. The results are established through a direct, analytic correspondence between transport coefficients for two component random media and the magnetization of the Ising model, which is based on the observation we made previously that both problems share the Lee-Yang property.