Theory Of Nonequilibrium Phase Transitions Research Articles

Unconventional criticality, scaling breakdown, and diverse universality classes in the Wilson-Cowan model of neural dynamics.

The Wilson-Cowan model constitutes a paradigmatic approach to understanding the collective dynamics of networks of excitatory and inhibitory units. It has been profusely used in the literature to analyze the possible phases of neural networks at a mean-field level, e.g., assuming large fully connected networks. Moreover, its stochastic counterpart allows one to study fluctuation-induced phenomena, such as avalanches. Here we revisit the stochastic Wilson-Cowan model paying special attention to the possible phase transitions between quiescent and active phases. We unveil eight possible types of such transitions, including continuous ones with scaling behavior belonging to known universality classes-such as directed percolation and tricritical directed percolation-as well as six distinct ones. In particular, we show that under some special circumstances, at a so-called "Hopf tricritical directed percolation" transition, rather unconventional behavior is observed, including the emergence of scaling breakdown. Other transitions are discontinuous and show different types of anomalies in scaling and/or exhibit mixed features of continuous and discontinuous transitions. These results broaden our knowledge of the possible types of critical behavior in networks of excitatory and inhibitory units and are, thus, of relevance to understanding avalanche dynamics in actual neuronal recordings. From a more general perspective, these results help extend the theory of nonequilibrium phase transitions into quiescent or absorbing states.

Dynamics of shock waves in bubble zones of finite size with a hydrate-forming gas

The purpose of this study is to study the dynamics of the wave field, which is realized in a channel with a liquid containing a rectangular zone with bubbles of the freon-12 hydrate-forming gas during the propagation of a pressure shock wave. In the initial state, the considered gas-liquid system is under pressure P0. After a sudden increase in pressure to the value of Pe, a pressure wave of a stepped profile propagates in the system and, as a result of the presence of a bubble curtain, its amplitude increases, which in turn has a more favorable effect on the formation of hydrate in gas bubbles. In the initial state, the hydrate formation process was not taken into account. As a result, the dynamics of the pressure wave is shown during its propagation in a semi-infinite channel containing a gas curtain with a hydrate-forming gas. The mechanism of gas hydrate formation is described in this work on the basis of the theory of nonequilibrium phase transitions in vapor-liquid systems.

Multi-dimensional complex phase transition model by catastrophe theory developing Kolmogorov turbulence theory

In this paper, a novel multi-dimensional complex non-equilibrium phase transition model is put forward to describe quantitatively the physical development process of turbulence and develop the Kolmogorov turbulence theory from the catastrophe theory, in which the well-known −5/3 power law is only a special case in this paper proving the accuracy of our methods. Catastrophe theory is a highly generalized mathematical tool that summarizes the laws of non-equilibrium phase transition. Every control variable in catastrophe theory could be skillfully expanded into multi-parameter multiplication with different indices and the relationship among these characteristic indices can be determined by dimensionless analysis. Thus, the state variables can be expressed quantitatively with multiple parameters, and the multi-dimensional non-equilibrium phase transition theory is established. As an example, by adopting the folding catastrophe model, we strictly derive out the quantitative relationship between energy and wave number with respect to a new scale index [Formula: see text] to quantitative study the whole process of the laminar flow to turbulence, in which [Formula: see text] varies from [Formula: see text] to [Formula: see text] corresponding to energy containing range and [Formula: see text] to energy containing scale where [Formula: see text] power law is deduced, and at [Formula: see text] the [Formula: see text] law of Kolmogorov turbulence theory is obtained, and fully developed turbulence phase starts at [Formula: see text] giving [Formula: see text] law. Furthermore, this theory presented is verified by our wind tunnel experiments. This novel non-equilibrium phase transition methods cannot only provide a new insight into the turbulence model, but also be applied to other non-equilibrium phase transitions.

Dynamics of longitudinal magnetization in transverse-field quantum Ising model: from symmetry-breaking gap to Kibble–Zurek mechanism

We show that the symmetry-breaking gap of the quantum Ising model in the transverse field can be extracted from free evolution of the longitudinal magnetization taking place after a gradual quench of the magnetic field. We perform for this purpose numerical simulations of the Ising chains with either periodic or open boundaries. We also study the condition for adiabaticity of evolution of the longitudinal magnetization finding excellent agreement between our simulations and the prediction based on the Kibble–Zurek theory of non-equilibrium phase transitions. Our results should be relevant for ongoing cold atom and ion experiments targeting either equilibrium or dynamical aspects of quantum phase transitions. They could be also useful for benchmarking D-Wave machines.

Rayleigh–Benard Instability: a Study by the Methods of Cahn–Hillard Theory of Nonequilibrium Phase Transitions

This article is an attempt to study the process of Rayleigh–Benard convective instability by the methods used for mathematical modeling of critical phenomena as nonequilibrium phase transitions in their initial stages of spinodal decomposition. We show that it is possible to extend the formalism adopted in the Cahn–Hillard theory of nonequilibrium phase transitions and perfected on problems of highgradient crystallization to other types of problems, in particular, those pertaining to the Rayleigh–Benard convective instability. For the initial stage of instability, a model is constructed that represents it as a nonequilibrium phase transition due to diffusive stratification. It is shown that the Gibbs free energy of deviation from the homogeneous state (with respect to the instability under consideration) is an analogue of the Ginsburg–Landau potential. Numerical experiments, by means of boundary temperature control, have been conducted with regard to self-excitation of the homogeneous state. Numerical analysis shows that convective flows may appear and proceed from regular forms (the so-called regular structures) to nonregular flows through a chaotization of the process. External factors, such as temperature growth, may lead to chaos via period doubling bifurcations.

A quantitative nonequilibrium phase transition theory for analyzing the turbulence development process

In this paper, a quantitative nonequilibrium multi-dimensional phase transition theory is proposed for describing the turbulence spectrum (energy E with wave number k and scaling index [Formula: see text]) of the turbulence development process by a fold catastrophe model. Each of the control variables in this catastrophe model is subtly expressed into a relative multi-parameter multiplication, and then the state variable can be quantitatively described by these parameters. By using this nonequilibrium phase transition theory, the quantitative relationship in the process of turbulence formation can be strictly derived through dimensionless analysis. Therefore, the turbulence development process can be described with respect to a scaling index [Formula: see text], in which there exists an energy containing range with −1.12 power law (E [Formula: see text] k[Formula: see text]) when [Formula: see text] varies from −2 to −1.2, and an inertial subrange with −1.69 power law (E [Formula: see text] k[Formula: see text]) that is almost identical with the famous Kolmogorov’s −5/3 power law when [Formula: see text] varies from −1.2 to −0.8, and then the dissipation range with −2.52 power law (E [Formula: see text] k[Formula: see text]) when [Formula: see text] varies from −0.8 to 0. Furthermore, this quantitative nonequilibrium phase transition theory has been verified by the corresponding theoretical comparison and experiment. This theory provides not only a new understanding of turbulence, but also a new perspective for other complex nonequilibrium phase transitions.

One-half of the Kibble–Zurek quench followed by free evolution

We drive the one-dimensional quantum Ising chain in the transverse field from the paramagnetic phase to the critical point and study its free evolution there. We analyze excitation of such a system at the critical point and dynamics of its transverse magnetization and Loschmidt echo during free evolution. We discuss how the system size and quench-induced scaling relations from the Kibble–Zurek theory of non-equilibrium phase transitions are encoded in quasi-periodic time evolution of the transverse magnetization and Loschmidt echo.

Spatial Kibble–Zurek mechanism through susceptibilities: the inhomogeneous quantum Ising model case

We study the quantum Ising model in the transverse inhomogeneous magnetic field. Such a system can be approached numerically through exact diagonalization and analytically through the renormalization group techniques. Basic insight into its physics, however, can be obtained by adopting the Kibble–Zurek theory of non-equilibrium phase transitions to description of spatially inhomogeneous systems at equilibrium. We employ all these approaches and focus on derivatives of longitudinal and transverse magnetizations, which have extrema near the critical point. We discuss how these extrema can be used for locating the critical point and for verification of the Kibble–Zurek scaling predictions in the spatial quench.

Critical behavior in earthquake energy dissipation

We explore bursty multiscale energy dissipation from earthquakes flanked by latitudes 29° S and 35.5° S, and longitudes 69.501° W and 73.944° W (in the Chilean central zone). Our work compares the predictions of a theory of nonequilibrium phase transitions with nonstandard statistical signatures of earthquake complex scaling behaviors. For temporal scales less than 84 hours, time development of earthquake radiated energy activity follows an algebraic arrangement consistent with estimates from the theory of nonequilibrium phase transitions. There are no characteristic scales for probability distributions of sizes and lifetimes of the activity bursts in the scaling region. The power-law exponents describing the probability distributions suggest that the main energy dissipation takes place due to largest bursts of activity, such as major earthquakes, as opposed to smaller activations which contribute less significantly though they have greater relative occurrence. The results obtained provide statistical evidence that earthquake energy dissipation mechanisms are essentially “scale-free”, displaying statistical and dynamical self-similarity. Our results provide some evidence that earthquake radiated energy and directed percolation belong to a similar universality class.

Tsallis non-extensive statistics and solar wind plasma complexity

This article presents novel results revealing non-equilibrium phase transition processes in the solar wind plasma during a strong shock event, which took place on 26th September 2011. Solar wind plasma is a typical case of stochastic spatiotemporal distribution of physical state variables such as force fields (B→,E→) and matter fields (particle and current densities or bulk plasma distributions). This study shows clearly the non-extensive and non-Gaussian character of the solar wind plasma and the existence of multi-scale strong correlations from the microscopic to the macroscopic level. It also underlines the inefficiency of classical magneto–hydro-dynamic (MHD) or plasma statistical theories, based on the classical central limit theorem (CLT), to explain the complexity of the solar wind dynamics, since these theories include smooth and differentiable spatial–temporal functions (MHD theory) or Gaussian statistics (Boltzmann–Maxwell statistical mechanics). On the contrary, the results of this study indicate the presence of non-Gaussian non-extensive statistics with heavy tails probability distribution functions, which are related to the q-extension of CLT. Finally, the results of this study can be understood in the framework of modern theoretical concepts such as non-extensive statistical mechanics (Tsallis, 2009), fractal topology (Zelenyi and Milovanov, 2004), turbulence theory (Frisch, 1996), strange dynamics (Zaslavsky, 2002), percolation theory (Milovanov, 1997), anomalous diffusion theory and anomalous transport theory (Milovanov, 2001), fractional dynamics (Tarasov, 2013) and non-equilibrium phase transition theory (Chang, 1992).

Understanding bursty behavior in midlatitude geomagnetic activity

We provide a new example of a natural system exhibiting statistical signatures of complex scaling behavior predicted by the contemporary theory of nonequilibrium phase transitions. We examine statistical properties of bursty multiscale energy dissipation in the inner magnetosphere of Earth based on the dynamics of the SYM‐H index, a global marker of low‐latitude geomagnetic fluctuations. We show that on average, and for time scales shorter than 2 h, temporal development of SYM‐H bursts follows an algebraic form consistent with the predictions from the theory of nonequilibrium phase transitions. Probability distributions of sizes and lifetimes of the activity bursts reveal no characteristic scales other than the scales imposed by technical limitations of the analysis. This behavior is observed for a wide range of SYM‐H burst durations starting from about 5 min up to 10–15 days. The power law exponents describing the probability distributions suggest that the main energy dissipation in the inner magnetosphere takes place because of large activity bursts such as major space storms as opposed to smaller activations whose contribution is less significant despite their much higher relative occurrence. The results obtained provide statistical evidence that the energy dissipation mechanisms associated with magnetospheric activity in the inner magnetosphere are essentially “scale‐free,” displaying dynamical and statistical self‐similarity. Our results can also be used for validating existing and future ring current models in terms of their ability to correctly represent the cross‐scale coupling effects in this system. They show what could be the first quantitative evidence for the same universality class as directed percolation in a natural system.

Macrokinetics Hierarchies of States at Dynamic Superplasticity

Dynamic superplasticity of aluminium alloys is considered from positions of the theory of nonequilibrium phase transitions. Evolution of the open nonequilibrium system by which temperature-rates process in materials is modelled, is investigated within the framework of worked out defining relations with attraction of thermodynamic functions of the response. It is shown, that to peak of superplasticity there corresponds a maximum of a specific thermal capacity. The entropy production in optimum thermomechanical conditions of superplasticity aspires to a minimum which corresponds to formation of qualitative ultrafine-grained structure. With use of the Focker-Plank equation mechanisms of deformation characteristic for superplasticity and boundary conditions are analyzed. It is shown, that at superplasticity the main is the mechanism grain-boundary sliding, and in metastable conditions are added diffusionary processes. Influence of the specified processes in conditions of superplasticity becomes prevailing.

Power Spectra of the Total Occupancy in the Totally Asymmetric Simple Exclusion Process

As a solvable and broadly applicable model system, the totally asymmetric exclusion process enjoys iconic status in the theory of nonequilibrium phase transitions. Here, we focus on the time dependence of the total number of particles on a 1-dimensional open lattice and its power spectrum. Using both Monte Carlo simulations and analytic methods, we explore its behavior in different characteristic regimes. In the maximal current phase and on the coexistence line (between high and low density phases), the power spectrum displays algebraic decay, with exponents -1.62 and -2.00, respectively. Deep within the high and low density phases, we find pronounced oscillations, which damp into power laws. This behavior can be understood in terms of driven biased diffusion with conserved noise in the bulk.

Metastability, criticality and phase transitions in brain and its models

This survey of experimental findings and theoretical insights of the past 25 years places the brain firmly into the conceptual framework of nonlinear dynamics, operating at the brink of criticality, which is achieved and maintained by self-organization. It is here the basis for proposing that the application of the twin concepts of scaling and universality of the theory of non-equilibrium phase transitions can serve as an informative approach for elucidating the nature of underlying neural-mechanisms, with emphasis on the dynamics of recursively reentrant activity flow in intracortical and cortico-subcortical neuronal loops.

A nonequilibrium phase transition in immune response

The dynamics of immune response correlated to signal transduction in immune thymic cells (T cells) is studied. In particular, the problem of the phosphorylation of the immune-receptor tyrosine-based activation motifs (ITAM) is explored. A nonlinear model is established on the basis of experimental observations. The behaviours of the model can be well analysed using the concepts of nonequilibrium phase transitions. In addition, the Riemann–Hugoniot cusp catastrophe is demonstrated by the model. Due to the application of the theory of nonequilibrium phase transitions, the biological phenomena can be clarified more precisely. The results can also be used to further explain the signal transduction and signal discrimination of an important type of immune T cell.

Kinetic equation in the theory of nonequilibrium phase transitions induced by ?colored? multiplicative noise

The formalism presented in this paper for deriving a kinetic equation for a dynamical system with multiplicative colored noise is to quite a degree universal. It can be used to investigate nonequilibrium phase transitions induced by both additive and multiplicative colored noise. The formalism can be used to describe, the behavior of a system under the influence of several stationary noise sources. The formalism can be generalized to the case of a system of equations with noise sources. One can investigate in its framework systems of the form\(\dot x = f_0 (x, t) + \eta (t)f_1 (x, t)\),i.e., systems subject simultaneously to noise and a deterministic force.

New problems in the theory of nonequilibrium phase transitions

New problems in the theory of nonequilibrium phase transitions

Novye zadachi v teorii neravnovesnykh fazovykh perekhodov

Novye zadachi v teorii neravnovesnykh fazovykh perekhodov

Stochastic theory of nonequilibrium phase transitions: application to phospholipid bilayer membranes

Etude des transitions de phase dans les couches bimoleculaires de phosphatidylcholine en fonction de la temperature

The systematic adiabatic elimination of fast variables from a many-dimensional Fokker-Planck equation

The Chapman-Enskog method for the adiabatic elimination of fast variables is applied to a general Fokker-Planck equation linear in the fast variables. This equation is the counterpart of the generalized Haken-Zwanzig model, a system of coupled Langevin equations often encountered in quantum optics and in the theory of non-equilibrium phase transitions. After a few equilibration times for the fast variables the time dependence of smooth solutions of this Fokker-Planck equation is completely governed by the reduced distribution function of the slow variables, which obeys a closed evolution equation. We obtain an explicit perturbation series for the generator of this reduced evolution. The system considered here is the most general system for which the Chapman-Enskog hierarchy can be solved explicitly by exploiting an analogy between the unperturbed operator and the Hamiltonian for coupled harmonic oscillators.