Out-of-equilibrium environment

Equation of Motion of a Particle Linearly Coupled to an Out-of-Equilibrium Environment... 

In Section VI, we consider a classical particle diffusing in an out-of-equilibrium environment. In this case, all the dynamical variables attached to the particle, even its velocity, are aging variables. We analyze how the drift and diffusion properties of the particle can be interpreted in terms of an effective temperature of the medium. From an experimental point of view, independent measurements of the mean-square displacement and of the mobility of a particle immersed in an aging medium such as a colloidal glass give access to an out-of-equilibrium generalized Stokes-Einstein relation, from which the effective temperature of the medium can eventually be deduced. 

The fully general situation of a particle diffusing in an out-of-equilibrium environment is much more difficult to describe. Except for the particular case of a stationary environment, the motion of the diffusing particle cannot be described by the generalized Langevin equation (22). A more general equation of motion has to be used. The fluctuation-dissipation theorems are a fortiori not valid. However, one can try to extend these relations with the help of an age- and frequency-dependent effective temperature, such as proposed and discussed, for instance, in Refs. 5 and 6. 

Let us now come back to the specific problem of the diffusion of a particle in an out-of-equilibrium environment. In a quasi-stationary regime, the particle velocity obeys the generalized Langevin equation (22). The generalized susceptibilities of interest are the particle mobility p(co) = Xvxi03) and the generalized friction coefficient y(co) = — (l/mm)x ( ) [the latter formula deriving from the relation (170) between y(f) and Xj> (f))- The results of linear response theory as applied to the particle velocity, namely the Kubo formula (156) and the Einstein relation (159), are not valid out-of-equilibrium. The same... 

In an out-of-equilibrium environment, no well-defined thermodynamical temperature does exist. Since, in an out-of-equilibrium regime, even if stationary, the FDTs are not satisfied, one can try to rewrite them in a modified way, and thus to extend linear response theory, with the help of a (frequency-dependent) effective temperature. 

The second law of thermodynamics asserts that the total entropy 5 of a system may change in time because of exchanges with its environment and internal entropy production which is vanishing at equilibrium and positive out of equilibrium [5]... 

We turn to Figure 6.2, which shows a specific mass, let us say 1 mol, of air that is out of equilibrium with the environment only with respect to its temperature T, which has been chosen to be larger than T0, the temperature of the environment. As has been shown before in this chapter, Equation 6.11, thermodynamics allows us to calculate the minimum amount of work required to bring this mass out of equilibrium with the environment Weired Weired... 

Physical exergy out of equilibrium with the environment in temperature. 

In Figure 6.2, we studied an arbitrary mass of air that was out of equilibrium with its environment because its temperature T > T0. We mentioned that thermodynamics allows us to calculate the minimum amount of work, Wr " / that it takes to create this nonequilibrium situation. Well considered, this amount of air creates a structure within its environment, in which a different... 

Recall that exergy values reflect the extent to which a compound or mixture is out of equilibrium with our environment. Examples are differences in pressure and temperature with the environment. Differences in temperature lead to heat transfer, while differences in pressure lead to mass flow. Chapter 6 shows that the physical exergy represents the maximum amount of work that can be obtained from a system by converting a system s pressure and temperature to those of our environment. 

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