Fluctuations Of Thermodynamic Variables Research Articles

Universality and scaling in homogeneous compressible turbulence

Turbulent flows, ubiquitous in nature and engineering, comprise fluctuations over a wide range of spatial and temporal scales. While flows with fluctuations in thermodynamic variables are much more common, much less is known about these flows than their incompressible counterparts in which thermodynamics is decoupled from hydrodynamics. A critical element in the study of the latter has been the concept of universal scaling laws which provides fundamental as well as practical information about the spatio-temporal behavior of these complex systems. Part of this success is due to the dependence on a single non-dimensional parameter, that is the Reynolds number. Universality in compressible flows, on the other hand, have proven to be more elusive as no unifying set of parameters were found to yield universal scaling laws. This severely limits our understanding of these flows and the successful development of theoretically sound models. Using results in specific asymptotic limits of the governing equations we show that universal scaling is indeed observed when the set of governing parameters is expanded to include internally generated dilatational scales which are the result of the driving mechanisms that produce the turbulence. The analysis demonstrate why previous scaling laws fail in the general case, and opens up new venues to identify physical processes of interest and create turbulence models needed for simulations of turbulent flows at realistic conditions. We support our results with a new massive database of highly-resolved direct numerical simulations along with all data available in the literature. In search of universal features, we suggest classes which bundle the evolution of flows in the new parameter space. An ultimate asymptotic regime predicted independently by renormalization group theories and statistical mechanics is also assssed with available data.

Fluctuation effects of thermodynamic variables in particle-spilling-from-well model with single potential well

Under ergodic condition, the analytic expressions for the fluctuations of temperature and particle number in the well of a system are obtained rigorously in particle-spilling-from-well model by means of integration approaches in phase space; the relationships among fluctuations of thermodynamic variables, the total particle numbers, and the well volume are emphasized. Results show that the less the total number of particles and the smaller the well's volume, the more remarkable the thermodynamic fluctuations effect, and the more closely the thermodynamic fluctuations related to the particle spilling from the well. There is a one to one correspondence between particle spilling from the well and the occurrence of negative specific heat, and the huge thermodynamic fluctuations,as well as, the origin of this phenomenon can be understood by the apparent energy inverse partition.

Fluctuations of thermodynamic variables in stationary compressible turbulence

AbstractA large database of new direct numerical simulations of forced compressible turbulence on up to$204{8}^{3} $grids, and a range of Reynolds (${R}_{\lambda } $) and turbulent Mach (${M}_{t} $) numbers, is analysed to study the scaling of pressure, density and temperature fluctuations. Small-perturbation analysis is used to study the scaling of variances, and different cross-correlations as well as spectra. Qualitative differences are observed between low and high${M}_{t} $. The probability density functions (p.d.f.s) of pressure and density are negatively skewed at low${M}_{t} $(consistent with incompressible results) but become positively skewed at high${M}_{t} $. The positive tails are found to follow a log-normal distribution. A new variable is introduced to quantify departures from isentropic fluctuations (an assumption commonly used in the literature) and is found to increase as${ M}_{t}^{2} $. However, positive fluctuations of pressure and density tend to be more isentropic than negative fluctuations. In general, Reynolds number effects on single-point statistics are observed to be weak. The spectral behaviour of pressure, density and temperature is also investigated. While at low${M}_{t} $, pressure appears to scale as${k}^{- 7/ 3} $($k$is the wavenumber) in the inertial range as in incompressible flows, a${k}^{- 5/ 3} $scaling also appears to be consistent with the data at a range of Mach numbers. Density and temperature spectra are found to scale as${k}^{- 5/ 3} $for a range of Mach numbers.

Statistical Distribution of a Completely Open System Based on Incomplete Shannon Entropy

Micro canonical, canonical and grand canonical systems are special cases of completely open systems. Within the framework of non-extensive and incomplete statistics, we derive the statistical distribution for a completely open system on the basis of incomplete Shannon entropy, using the maximum entropy principle. We calculate the physical properties of a linear filament using this distribution. The results are the same as those calculated using the incomplete E-V distribution, except that average values of the thermodynamic variables replace the corresponding constant values. However, the relative fluctuations in thermodynamic variables are completely different. In the canonical and grand canonical distributions, relative fluctuations are proportional to . In the incomplete statistical distribution for a completely open system, the relative fluctuations are proportional to 1. This result can explain some phenomenon that traditional and current statistical mechanics cannot explain, such as the large fluctuations of critical, super-cooled and overheated states.

Thermal conductivity and chiral critical point in heavy ion collisions

Background: Quantum chromodynamics is expected to have a phase transition in the same static universality class as the 3D Ising model and the liquid-gas phase transition. The properties of the equation of state, the transport coefficients, and especially the location of the critical point are under intense theoretical investigation. Some experiments are underway, and many more are planned, at high-energy heavy ion accelerators.Purpose: Develop a model of the thermal conductivity, which diverges at the critical point, and use it to study the impact of hydrodynamic fluctuations on observables in high-energy heavy ion collisions.Methods: We apply mode-coupling theory, together with a previously developed model of the free energy that incorporates the critical exponents and amplitudes, to construct a model of the thermal conductivity in the vicinity of the critical point. The effect of the thermal conductivity on correlation functions in heavy ion collisions is studied in a boost-invariant hydrodynamic model via fluctuations, or noise.Results: We find that the closer a thermodynamic trajectory comes to the critical point the greater is the magnitude of the fluctuations in thermodynamic variables and in the 2-particle correlation functions in momentum space.Conclusions: It may be possible to discern the existence of a critical point, its location, and thermodynamic and transport properties near to it in heavy ion collisions using the methods developed here.

Calculation of excess free energies of precipitates via direct thermodynamic integration across phase boundaries

We describe a technique for constraining macroscopic fluctuations in thermodynamic variables well-suited for Monte Carlo (MC) simulations of multiphase equilibria. In particular for multicomponent systems this amounts to a statistical ensemble that implements constraints on both the average composition as well as its fluctuations. The variance-constrained semi-grand-canonical (VC-SGC) ensemble allows for MC simulations, in which single-phase systems can be reversibly switched into multiphase equilibria allowing the calculation of excess free energies of precipitates of complex shapes by thermodynamic integration. The basic features as well as the scaling and convergence properties of this technique are demonstrated by an application to an Ising model. Finally, the VC-SGC MC simulation technique is used to calculate $\ensuremath{\alpha}/{\ensuremath{\alpha}}^{\ensuremath{'}}$ interface free energies in Fe-Cr alloys as a function of orientation and temperature taking into account configurational, vibrational, and structural degrees of freedom.

Interaction of isotropic turbulence with shock waves: effect of shock strength

As an extension of the authors' work on isotropic vortical turbulence interacting with a shock wave (Lee, Lele & Moin 1993), direct numerical simulation and linear analysis are performed for stronger shock waves to investigate the effects of the upstream shock-normal Mach number (M1). A shock-capturing scheme is developed to accurately simulate the unsteady interaction of turbulence with shock waves. Turbulence kinetic energy is amplified across the shock wave, and this amplification tends to saturate beyond M1 = 3.0. An existing controversy between experiments and theoretical predictions on length scale change is thoroughly investigated through the shock-capturing simulation: most turbulence length scales decrease across the shock, while the dissipation length scale (ρq3/ε) increases slightly for shock waves with M1<1.65. Fluctuations in thermodynamic variables behind the shock wave are nearly isentropic for M1<1.2, and deviate significantly from isentropy for the stronger shock waves, due to the entropy fluctuation generated through the interaction.

Ensemble Dependence of Fluctuations with Application to Machine Computations

The standard theory of fluctuations in thermodynamic variables in various ensembles is generalized to nonthermodynamic variables: e.g., the mean-square fluctuations of the kinetic energy $K$ in a classical microcanonical ensemble at fixed energy $E$ is given, for large systems, by $\frac{〈{(\ensuremath{\delta}K)}^{2}〉}{〈K〉=T[\frac{1\ensuremath{-}3}{2C})}$, where $T$ is the temperature (corresponding to the energy $E$) and $C$ is the specific heat per particle (in units of Boltzmann's constant). The general results may be expressed in terms of the asymptotic behavior of the Ursell functions in various ensembles. Applications are made to molecular dynamic computations where time averages correspond (via ergodicity) to phase averages in an ensemble with fixed energy and momentum. The results are also useful for time-dependent correlations.