Fluctuation-dissipation theorem thermodynamics

Fluctuation-dissipation theorem, transition state trajectory, white noise, 203—207 Fluctuation theorem, nonequilibrium thermodynamics, 6—7... 

G. N. Bochkov and Y. E. Kuzovlev, Non-linear fluctuation relations and stochastic models in nonequilibrium thermodynamics. 1. Generalized fluctuation-dissipation theorem. Physica A 106, 443-J79 (1981). 

Another advantage of the simulation is its abihty to make direct tests on the range of validity of basic thermodynamical theorems such as the fluctuation-dissipation theorem. In the second paper of the series by Evans, he considers these points for the simplest type of torque mentioned above, —X F. Consider the return to equilibrium of a dynamical variable A after taking off at r = 0 the constant torque appUed prior to this instant in time. If the torque is removed instantaneously, the first fluctuation-dissipation theorem implies that the normalized fall transient will decay with the same dependence as the autocorrelation function (A(t)A(O))- Al /(A 0)) — Therefore,... 

Molecular dynamics simulations entail integrating Newton s second law of motion for an ensemble of atoms in order to derive the thermodynamic and transport properties of the ensemble. The two most common approaches to predict thermal conductivities by means of molecular dynamics include the direct and the Green-Kubo methods. The direct method is a non-equilibrium molecular dynamics approach that simulates the experimental setup by imposing a temperature gradient across the simulation cell. The Green-Kubo method is an equilibrium molecular dynamics approach, in which the thermal conductivity is obtained from the heat current fluctuations by means of the fluctuation-dissipation theorem. Comparisons of both methods show that results obtained by either method are consistent with each other [55]. Studies have shown that molecular dynamics can predict the thermal conductivity of crystalline materials [24, 55-60], superlattices [10-12], silicon nanowires [7] and amorphous materials [61, 62]. Recently, non-equilibrium molecular dynamics was used to study the thermal conductivity of argon thin films, using a pair-wise Lennard-Jones interatomic potential [56]. 

When we consider the dependence of excess heat production on an external transformation, we can connect thermodynamic quantities and underlying dynamics. We have derived the theorem similar to the fluctuation-dissipation theorem [10]. The theorem shows that thermodynamic entropy production such as excess heat can be written as a correlation function between Einstein-Shanon entropy functions. Through the correlation function the thermodynamic entropy production is related to the underlying dynamics. 

In Section II we will review thermodynamics and the fluctuation-dissipation theorem for excess heat production based on the Boltzmann equilibrium distribution. We will also mention the nonequilibrium work relation by Jarzynski. In Section III, we will extend the fluctuation-dissipation theorem for the superstatisitcal equilibrium distribution. The fluctuation-dissipation theorem can be written as a superposition of correlation functions with different temperatures. When the decay constant of a correlation function depends on temperature, we can expect various behaviors in the excess heat. In Section IV, we will consider the case of the microcanonical equilibrium distribution. We will numerically show the breaking of nonergodic adiabatic invariant in the mixed phase space. In the last section, we will conclude and comment. 

A consequence of these equations is that the dissipative kinetic coefficient c can be related to the correlation coefficient (p(t), p(t + t )> via the fluctuation dissipation theorem [23, 24]. According to Equation (22), we can conclude that the fluctuations of thermodynamic fluxes are similar to an impressed macroscopic deviation, except they appear spontaneously. 

The fluctuations at thermodynamic equilibrium are related in a universal way to the kinetic response according to the fluctuation-dissipation theorem (FTD). 

Thus, the requirement that the Brownian particle becomes equilibrated with the surrounding fluid fixes the unknown value of A, and provides an expression for it in terms of the friction coefficient, the thermodynamic temperature of the fluid, and the mass of the Brownian particle. Equation (A3.1.63) is the simplest and best known example of a fluctuation-dissipation theorem, obtained by using an equilibrium condition to relate the strength of the fluctuations to the frictional forces acting on the particle [22],... 

Here T is the local-equilibrium temperature. In extended irreversible thermodynamics fluxes are independent variables. The kinetic temperature associated to the three spatial directions of along the flow, along the velocity gradient, and perpendicular to the previous to directions may be different from each other. To define temperature from the entropy is the most fundamental definition, and the nonequilibrium temperature may come from the derivative of a nonequilibrium entropy du/dS) -p. Effective nonequilibrium temperature may be defined from the fluctuation-dissipation theorem relating response function and correlation function. 

Fluctuation-dissipation theorem of linear nonequilibrium thermodynamics... 

Since chemical reaction is considered as a stochastic process, and furthermore as a thermodynamic process, it is a natural question to ask what are the counterparts of the statements of the fluctuation theory of nonequilibrium thermodynamics. In the theory of thermodynamics the fluctuation-dissipation theorem is associated with the observation that the dissipative process leading to equilibrium is connected with fluctuations around that equilibrium. This fact was pointed out in a particular case (related to Brownian movement) by Einstein. Different representations of the theorem exist for linear thermodynamic processes (Callen Welton, 1951 Greene Callen, 1951 Kubo 1957 Lax, 1960 van Vliet Fasset, 1965 van Kampen, 1965.)... 

FIGURE 1 Distribution of work values for many repetitions (realizations) of a thermodynamic process involving a nanoscale system. The process might involve the stretching of a single RNA molecule or, perhaps, the compression of a tiny quantity of gas by a microscopic piston. The tail to the left of AF represents apparent violations of the second law of thermodynamics. The distribution p( W) satisfies the nonequilibrium work theorem (Eq.6), which reduces to the fluctuation-dissipation theorem (Eq.l) in the special case of near-equUibrium processes. 

Before we come to these models, we will first introduce a basic law of statistical thermodynamics which we require for the subsequent treatments and this is the fluctuation-dissipation theorem . We learned in the previous chapter that the relaxation times showing up in time- or frequency dependent response functions equal certain characteristic times of the molecular dynamics in thermal equilibrium. This is true in the range of linear responses, where interactions with applied fields are always weak compared to the internal interaction potentials and therefore leave the times of motion unchanged. The fluctuation-dissipation theorem concerns this situation and describes explicitly the relation between the microscopic dynamics in thermal equilibration and macroscopic response functions. 

Imagine that we select within a sample a subsystem contained in a volume Vj which is small but still macroscopic in the sense that statistical thermodynamics can be applied. If we could measure the properties of this subsystem we would observe time dependent fluctuations, for example in the shape of the volume, i.e. the local strain, the internal energy, the total dipole moment, or the local stress. The fluctuation-dissipation theorem relates these spontaneous, thermally driven fluctuations to the response functions of the system. We formulate the relationship for two cases of interest, the fluctuations of the dipole moments in a polar sample and the fluctuations of stress in a melt. 

Callen Herbert Bernard (1919-1990) US. phys., research on solid-state physics, thermodynamics and statistical mechanics, fluctuation-dissipation theorem (book Thermodynamics introduction to thermostatics 1960)... 

According to the fluctuation-dissipation theorem the equilibrium mean square deviation of thermal composition fluctuations (5 ) is related to the first derivative of the order parameter with respect to the chemical potential [3,4]. If the order parameter is defined as the composition

conjugate field is represented by the difference of the chemical potentials Afi(= - mb) so the degree of thermal fluctuations is related to dAfi idA/x = d/dA/x) which is equivalent to 1/9 AG, where AG represents the Gibbs free energy of mixing with its natural parameters temperature T, pressure P, and composition This shows how the thermodynamic parameters can be determined from measurements of thermal fluctuations. 

A second catch is the noise. If one observes the movements of a colloidal particle, the Brownian motion will be evident. There may be a constant drift in the dynamics, but the movement will be irregular. Likewise, if one observes a phase-separating liquid mixture on the mesoscale, the concentration levels would not be steady, but fluctuating. The thermodynamic mean-field model neglects all fluctuations, but they can be restored in the dynamical equations, similar to added noise in particle Brownian dynamics models. The result is a set of stochastic diffusion equations, with an additional random noise source tj [20]. In principle, the value and spectrum of the noise is dictated by a fluctuation dissipation theorem, but usually one simply takes a white noise source. 

The minimum entropy production theorem dictates that, for a system near equilibrium to achieve a steady state, the entropy production must attain the least possible value compatible with the boundary conditions. Near equilibrium, if the steady state is perturbed by a small fluctuation (8), the stability of the steady state is assured if the time derivative of entropy production (P) is less than or equal to zero. This may be expressed mathematically as dPIdt 0. When this condition pertains, the system will develop a mechanism to damp the fluctuation and return to the initial state. The minimum entropy production theorem, however, may be viewed as providing an evolution criterion since it implies that a physical system open to fluxes will evolve until it reaches a steady state which is characterized by a minimal rate of dissipation of energy. Because a system on the thermodynamic branch is governed by the Onsager reciprocity relations and the theorem of minimum entropy production, it cannot evolve. Yet as a system is driven further away from equilibrium, an instability of the thermodynamic branch can occur and new structures can arise through the formation of dissipative structures which requires the constant dissipation of energy. 

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