Thermodynamic theory of equilibrium fluctuations

The postulational basis of classical thermodynamics has been expanded to incorporate equilibrium fluctuations. The main additional elements of the proposed thermodynamic theory are the concept of quasi-equilibrium states, a definition of non-equilibrium entropy, a fundamental equation of state in the entropy representation, and a fluctuation postulate describing the probability distribution of macroscopic parameters of an isolated system. Although these elements introduce a statistical component that does not exist in classical thermodynamics, the logical structure of the theory is different from that of statistical mechanics and represents an expanded version of thermodynamics. Based on this theory, we present a regular procedure for calculations of equilibrium fluctuations of extensive parameters, intensive parameters and densities in systems with any number of fluctuating parameters. The proposed fluctuation formalism is demonstrated by four applications: (1) derivation of the complete set of fluctuation relations for a simple fluid in three different ensembles; (2) fluctuations in finite-reservoir systems interpolating between the canonical and micro-canonical ensembles; (3) derivation of fluctuation relations for excess properties of grain boundaries in binary solid solutions, and (4) derivation of the grain boundary width distribution for pre-melted grain boundaries in alloys. The last two applications offer an efficient fluctuation-based approach to calculations of interface excess properties and extraction of the disjoining potential in pre-melted grain boundaries. Possible future extensions of the theory are outlined.     

Grain-boundary fluctuations in two-dimensional colloidal crystals

We study grain-boundary fluctuations in two-dimensional colloidal crystals in real space and time using video microscopy. The experimentally obtained static and dynamic correlation functions are very well described by expressions obtained using capillary wave theory. This directly leads to values for the interfacial stiffness and the interface mobility, the key parameters in curvature-driven grain-boundary migration. Furthermore, we show that the average grain-boundary position exhibits a one-dimensional random walk as recently suggested by computer simulations [Z. T. Trautt, M. Upmanyu, and A. Karma, Science 314, 632 (2006)]. The interface mobility determined from the mean-square displacement of the average grain-boundary position is in good agreement with values inferred from grain-boundary fluctuations.     

Nonthermodynamic fluctuations in two-dimensional hydrodynamic systems

The problem of fluctuations in two-dimensional hydrodynamic systems under nonequilibrium conditions due to external forces is studied on the example of a simple model. The effective action is given; this action makes it possible to construct a perturbation series to calculate fluctuation effects in such systems. Renormalizability of this action with an infinite number of vertices is proved; renormalization group equations are derived. A number of solutions of these equations are found; the asymptotic behaviour of the solution is considered in the limit of large logarithms.     

Magnetic-field-induced non-Gaussian fluctuations in macroscopic equilibrium systems

We calculate the magnetic-field-dependent nonlinear conductance and noise in a two-dimensional macroscopic inhomogeneous system. If the system does not possess a specific symmetry, the magnetic field induces a nonzero third cumulant of the current even at equilibrium. This cumulant is related to the first and second voltage derivatives of the spectral density and average current in the same way as for mesoscopic quantum-coherent systems, but these quantities may be much larger. The system provides a robust test of a nonequilibrium fluctuation relation.     

Roughness of crack interfaces in two-dimensional beam lattices

The roughness of crack interfaces is reported in quasistatic fracture, using an elastic network of beams with random breaking thresholds. For strong disorders we obtain zeta = 0.86(3) for the roughness exponent, a result which is very different from the minimum energy surface exponent, i.e., zeta = 2 / 3. A crossover to lower values is observed as the disorder is reduced, the exponent in this regime being strongly dependent on the disorder.     

Density fluctuations of two-dimensional active-passive mixtures

Dimension-dependent giant density fluctuations are a typical feature of active matter systems. In this work, we study the density fluctuation in two-dimensional mixtures of active and passive particles by Brownian dynamics simulations. The boundary of motility-induced phase separation is determined by the transition from unimodal to bimodal density distribution. A rapid increase of the fluctuation exponent near the boundary of phase separation in the plane of density and Péclet number was observed. When phase separation occurs, the fluctuation exponent is an approximate constant of $0.85.$     

The shapes of bowed interfaces in the two-dimensional Ising model

We show that properly normalized net energy fluctuations associated with interfaces in two-dimensional Ising models are described, asymptotically, by random walk partition functions. Two examples are investigated: one is a droplet on a wall, and the other is two nearby, ideally parallel interfaces; the mean shapes of the interfaces in both cases prove to be elliptic, bowed outward from the wall or from each other, the semiminor axis of the latter ellipse being 1/2 that of the former, in accord with random walk results.     

Thermodynamic fluctuations in two-dimensional degenerate antiferromagnetic structures

A model of a two-dimensional antiferromagnet with an arbitrary anisotropic interaction that allows for degeneracy of the ground state is proposed. The lifting of degeneracy by thermodynamic fluctuations and the accompanying effects are studied by a method of self-consistent calculations of Gaussian angular fluctuations that is asymptotically exact at low temperatures. Fluctuations are shown to lead to collinear ordering of the orientations of magnetic sublattices, an effect that initiates long-range orientational order in systems with anisotropic interaction but retains only short-range order in systems with isotropic short-range interaction. The temperature patterns of the orientational correlators are given for the particular cases of dipole and isotropic short-range interaction models. The nature of the Ising-like behavior of the system is discussed for the case of a strong anisotropy of the correlators, which corresponds to quasi-one-dimensional behavior. Zh. éksp. Teor. Fiz. 111, 669–680 (February 1997)     

Density fluctuations in strongly stratified two-dimensional turbulence

We present a study of density fluctuations in two-dimensional soap-film convection. When the temperature difference (DeltaT) imposed vertically along the film is smaller than a critical value (DeltaT(C) approximately 48 K), the convective motion is in a strongly stratified state and the frequency power spectrum of density fluctuations shows a Bolgiano-like scaling f(-7/5) in the buoyancy subrange. When DeltaT>DeltaT(C), the fluid motion crosses over to a strongly mixed state characterized by the emergence of a large-scale circulation. The density power spectrum in this state has a passive-scalar-like scaling f(-1.0).     

Shock fluctuations in the two-dimensional asymmetric simple exclusion process

We study via computer simulations (using various serial and parallel updating techniques) the time evolution of shocks, particularly the shock width(t), in several versions of the two-dimensional asymmetric simple exclusion process (ASEP). The basic dynamics of this process consists of particles jumping independently to empty neighboring lattice sites with ratesp _up=p _down=p andp _left<p _right. If the system is initially divided into two regions with densities _left< _right, the boundary between the two regions corresponds to a shock front. Macroscopically the shock remains sharp and moves with a constant velocityv _shock=(p _right– _left)(1–p _left–p _right). We find that microscopic fluctuations cause to grow ast , 1/4. This is consistent with theoretical expectations. We also study the nonequilibrium stationary states of the ASEP on a periodic lattice, where we break translation invariance by reducing the jump rates across the bonds between two neighboring columns of the system by a factorr. We find that for fixed overall density _avg and reduction factorr sufficiently small (depending on _avg and the jump rates) the system segregates into two regions with densities ₁ and ₂=1– ₁, where these densities do not depend on the overall density _avg. The boundary between the two regions is again macroscopically sharp. We examine the shock width and the variance in the shock position in the stationary state, paying particular attention to the scaling of these quantities with system size. This scaling behavior shows many of the same features as the time-dependent scaling discussed above, providing an alternate determination of the result1/4.     

Long-range correlations in two-dimensional spatio-temporal seismic fluctuations

We analysed the scaling behaviour of the two-dimensional (2-D) sequence (Δs, Δt) of the 1981–1998 southern California seismicity, where Δs is the distance between two consecutive earthquakes (jump) and Δt is their interevent interval. The 2-D seismic spatio-temporal fluctuations were investigated by means of the detrended fluctuation analysis (DFA), well-known methodology used to detect scaling behaviour in observational time series possibly affected by nonstationarities. The estimated scaling exponents α_DFA, larger than 0.5, indicate the presence of persistent long-range correlations in the 2-D sequence analysed. The variation of the scaling exponent with the increase of threshold magnitude shows a two-fold behaviour: in the range between 1.5 (the completeness magnitude of the catalog) and 3.0, the scaling exponent is quite constant and denoting a flicker-noise dynamics; while for magnitudes larger than 3.0 it decreases with the increase of magnitude, indicating a tendency toward a 2-D space–time Poissonian process for large events.     

Nonstatistical temperature fluctuations and displacement of equilibrium

It is shown that an anomalously large displacement of equilibrium occurs in systems with rapidly oscillating temperature. The thermodynamic parameters of such a system differ substantially from the values corresponding to constant temperature. The dynamics of the distributions of hydrogen ions and impurity helium and carbon ions over ionization states in a hydrogen plasma and the dynamics of the intensity distribution of the line radiation of impurity carbon in a hydrogen plasma with oscillating temperature are calculated as an example. In systems with fluctuating temperature, the effect in question could be important in virtually all problems based on levelwise kinetics, for example, the energy balance and spectral diagnostics of turbulent plasma, the criterion of thermal stability, the dynamics of turbulent plasma and gas jets, the conditions for amplifier and oscillator operation in the active media of electric-discharge gas lasers, and so on. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 5, 308–312 (10 March 1998)     

No anomalous fluctuations exist in stable equilibrium systems

A general theorem is rigorously proved for the case, when an observable is a sum of linearly independent terms: the dispersion of a global observable is normal if and only if all partial dispersions of its terms are normal, and it is anomalous if and only if at least one of the partial dispersions is anomalous. This theorem, in particular, rules out the possibility that in a stable system with Bose–Einstein condensate some fluctuations of either condensed or non-condensed particles could be anomalous. The conclusion is valid for arbitrary systems, whether uniform or non-uniform, interacting weakly or strongly. The origin of fictitious fluctuation anomalies, arising in some calculations, is elucidated.