Thermodynamic fluxes

In order to obtain a qualitative view of how the transition regime differs from the continuum flow or the slip flow regime, it is instructive to consider a system close to thermodynamic equilibrium. In such a system, small deviations from the equilibrium state, described by thermodynamic forces X, cause thermodynamic fluxes J- which are linear functions of the (see, e.g., [15]) ... 

The problems with the adiabatic Yamada-Kawasaki distribution and its thermostatted versions can be avoided by developing a nonequilibrium phase space probability distribution for the present case of mechanical work that is analogous to the one developed in Section IVA for thermodynamic fluxes due to imposed thermodynamic gradients. The odd work is required. To obtain this, one extends the work path into the future by making it even about t ... 

In the macroscopic theory, a state of a physical (chemical) system is described by a set of thermodynamic parameters ,-,/= 1,..., n. These parameters and their derivatives determine the values of the thermodynamic fluxes/,- and forces Xj, i= 1,.. 

The description of coupled flow and transport phenomena is usually based on non-equilibrium thermodynamics [2], Application of this theory leads to a set of linear equations, relating all thermodynamical fluxes Jj to all thermodynamical forces Xj in a system ... 

Expression for production of entropy (8.18) can be now compared with the general results of non-equilibrium thermodynamics, which are known for both non-stationary and stationary cases. It is obvious, that last term in the right-hand side of relation (8.18) corresponds to a non-stationary case and includes the equation of change of internal variables that is relaxation equation. The first two terms in formula (8.18) correspond to a stationary case and should be considered as the products of thermodynamic fluxes and thermodynamic forces (it is possible with any multipliers). When the internal variables are absent, we should write a relation between the fluxes and forces in the form... 

By the middle 2000s, the model used [42] had been physically and numerically enhanced by the introduction of biharmonic horizontal mixing of the momentum, free sea surface, and actual thermodynamic fluxes at all the open boundaries implemented with a 15-km horizontal resolution, 44 levels over the vertical and a 5-min time step [44,45]. In the latter papers, instead of the density fields [9], climatic temperature and salinity fields with a twice coarser horizontal resolution (about 37 km) were used based on a twofold greater database (about 100 000 stations). 

Far from thermodynamic equiHbrium we find nonfinear interdependence of thermodynamic fluxes and forces. In this case, the Onsager reciprocal relations are generally not satisfied, and the formafism developed in Chapter 2 is not fuUy applicable for analysis of the state of open systems. Analysis of systems that are far from thermodynamic equilibrium is the subject of nonlinear nonequilibrium thermodynamics. 

A typical problem in thermodynamics of systems that are far from their equilibrium is the analysis of the stability of stationary states of the system. Thermodynamic criteria of the stability of stationary states are found the same way as for systems that are far from and close to thermodynamic equilibrium (see Section 2.4) by analyzing signs of thermodynamic fluxes and forces arising upon infinitesimal deviation of the system from the inspected stationary state. If the system is in the stable stationary state, then any infinitesimal deviation from this state must induce the forces that push it to return to the initial position. 

Figure 3. A complete ystematization of modem thermodynamics. Note (AG )t.p, (AG2)t.p and (AG)t.p are Gibbs free energy changes of reaction 1, reaction 2 and the whole system at constant temperature and pressure, respectively. X and J represent thermodynamic force and thermodynamic flux for irreversible process, respectively.

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

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. 

Thus, the flux-force equations linearly related n thermodynamics fluxes Jy. Ji,.. Ao n conjugate external thermodynamic forces Fi. Fi-- - with the proportionality constants Lij being the n transport coefficients for the process. Note that Eq. (A.9) is an example of Eq. (A. 10) for n = 1 with Ji=J,Lii — a, and Fi—E. Thermoelectric conduction and thermal diffusion are examples of processes that obey Eq. (A. 10) for n > 1. 

Notice that Eq. (A.l la) is identical in form to Eq. (A.IO) so long as one interprets the A(t) s, the X(f) s, and the L s in Eq. (A.lla) as, respectively, the thermodynamic fluxes, forces, and transport coefficients for the process r Tp. However, while the F s in Eq. (A.IO) are external applied forces, the X s in Eq. (A.lla) are internal thermodynamic driving forces. Thus, in adopting Eq. (A.lla), Onsager has boldly postulated that the linear flux-force relations experimentally established for external forces hold as well for internal forces. 

Wherein the definition of the thermodynamic fluxes and forces of (A3,2,13) and (A3,2,14) have been used. Onsager defined [5] the analogue of the Rayleigh dissipation function by... 

The second, more general, treatment is based on nonequilibrium thermodynamics. Fluxes and forces are connected by a matrix. The diagonal elements (the main effects) of this matrix are well-known - for example, the diffusion coefficient (which is the connection between a particle flux under a concentration gradient) or the thermal conductivity (which relates the temperature gradient with the heat flux). One of the non-diagonal elements is the Seebeck coefficient (= thermopower, ]), which relates a temperature gradient with a particle flux. Based on this, general equations are obtained that describe the heat and particle flow in a thermal and concentration profile ... 

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. 

The thermodynamic force A, the conjuj te variable to the thermodynamic flux F, is "deflnaf as ... 

In many physical problems, the total dissipation can be written as the sum of the products between a set of thermodynamic forces X and theirs conjugates, the thermodynamic flux x ... 

CD inequality will no longer be automatically verified. This means that thermodynamic principles may then be violated in some evolutions. Note that in order to describe isotropac and kinematic hardening, the thermodynamic flux is often decomposed into a tensor a and a scalar r, associated with thermodynamic forces X and R. We would then have to write ... 

Before introducing the notion of nonequilibrium thermodynamics we shall first summarize briefly the linear and nonlinear laws between thermodynamic fluxes and forces. A key concept when describing an irreversible process is the macroscopic state parameter of an adiabatically isolated system These parameters are denoted by. At equilibrium the state parameters have values A , while an arbitrary state which is near or far from the equilibrium may be specified by the deviations from the equilibrium state ... 

In the original form of Onsager s relations, the vector of thermodynamic fluxes J and the vector of thermodynamic forces F are connected by a matrix L that is symmetric in the standard scalar product ... 

Gibbs free energy change of reaction (J moP ) enthalpy change of reaction (J moP ) thermodynamic flux (mol s ) equilibrium coefficient (depends) k column vector of kinetic coefficients (first order) (s )... 

In the crack layer theory (75,76) the movement of the crack and surrounding damage is decomposed into elementary movements translation, rotation, isotropic expansion, and distortion. In this way, the damage surrounding the tip can evolve in a general sense. These elementary movements become thermodynamic fluxes. The reciprocal forces contain an active part (energy release rates associated with each movement) Aj, and a resistive part (an energy barrier) Rt. Within the context of classical irreversible thermodynamics, the entropy of the system. Si, can be expressed in terms of a bilinear form of forces and fluxes shown in equation 27. 

There are NEMD schemes where cause and effect are reversed the flux is imposed and the corresponding field is measured. Gonceptually this approach is based on the theorems of Norton and Thevenin which are an example of the macroscopic duality that exists between what are generally recognized as thermodynamic fluxes and thermodynamic forces. The principle of reverse perturbation has been used by Miiller-Plathe for calculating the shear viscosity, thermal conductivity, Soret coefficients, and thermal diffusion. For example, in the calculation of shear viscosity, the reverse nonequilibrium molecular dynamics (rNEMD) method involves a simple exchange of particle momenta the effect, the... 

The X s depend on the thermodynamic fluxes. They can also depend on the intensive thermodynamic scalar variables Fj. We accordingly make the constitutive assumptions... 

The thermodynamic fluxes and the entropy production vanish at thermodynamic equilibrium... 

There are two further important properties of the thermodynamic flux potential [4], namely... 

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