Global Journal of Science Frontier Research, A: Physics and Space Science, Volume 23 Issue 1

contains definite forces X i and fluxes J i . This bestows it with a definite sense corresponding to an energy field strength Θ ik and averaged pulse k th energy source P i = Θ i υ i. Furthermore, it does not require a compilation of complex and cumbersome equations for the balance of matter, charge, momentum, energy and entropy. This dramatically simplifies the ability for thermodynamics to solve certain problems. III. T hermodynamic D erivation of the O nsager R eciprocal R elations One of the most important provisions of the theory of irreversible processes is the “reciprocity relation" L ij = L ji between the off-diagonal coefficients L ij and L i in the "phenomenological" laws postulated by L. Onsager: J i = Σ j L ij X j (9) These ratios establish the relationship between dissimilar fluxes J i and forces X j and reduce the number of proportionality coefficients between them to be experimentally determined from n 2 to n ( n +1)/2. To prove these relations, the future Nobel laureate L. Onsager had to use the theory of fluctuations, the principle of microscopic reversibility and an additional postulate about the linear nature of the laws of decay of fluctuations [1]. All three of these assumptions are somewhat outside of classical thermodynamics; therefore, he rightly called his theory "quasi- thermodynamics". Meanwhile, it can be shown that these relations gain support from the law of the conservation of energy (6). From that law, based on the independence of the mixed derivative from the order of differentiation with respect to the variables X i and X j ( i , j= 1,2,... n ), it follows: ∂ 2 U / ∂X i ∂X j = ∂ 2 U / ∂X j ∂X i (10) This directly implies the relationship between unlike fluxes and forces, which we term differential reciprocal relations [13] : ( ∂J i / ∂X j ) = ( ∂J j / ∂X j ). (11) These relations are applicable to both linear and nonlinear transport laws and allow any dependence of the coefficients L ij on the parameters of the equilibrium state ψ i and Θ i . Application to the linear laws (9) directly leads to the symmetry of the matrix of phenomenological coefficients L ij = L ji : ( ∂J i / ∂X j ) = L ij = ( ∂J j / ∂X j ) = L ji (12) Their derivation shows that these relationships are a consequence of more general reasons than the reversibility in time of micro processes. This explains why these relationships have often turned out to be valid far in domains far beyond the above conditions. IV. F inding " S uperposition E ffects" without using O nsager R elations In isolated systems, the sum of internal forces Σ i F i ( i = 1, 2… n ) is always zero. This means that, in accordance with Newton's 3 rd law, any one of them can be expressed as the sum of n =1 different forces of the j th kind: F i = - Σ n -1 F j . The relationship of these forces to thermodynamic forces X i is easy to construct. From the expression for power dW/dt = X i · J i = F i · i υ it follows that X i = F i / Θ i , i.e., it represents the precise meaning of force in its more general physical interpretation. Taking this into account, laws (17) can be represented in a form closer to (9): J i =L i Σ j Θ j X j , (13) Such a form of the laws of transfer and relaxation does not require the empirical coefficients L i to be constant; this expands the scope of these equations' applicability to nonlinear systems and states far from equilibrium. In addition, it allows to propose a new method for finding the "superposition effects" of irreversible process which are due to "partial" (incomplete) equilibria of the i th kind ( J i = 0). The specificity of this method is easier to understand with the example of the diffusion of the k th substance in a continuous heterogeneous composition (with the concentration of components c j , temperature T and pressure p ). According to laws (13), this process has the form: J k = – D k μ k , (14) where D k is the diffusion coefficient of the k th substance; μ k is its chemical potential. If we represent μ k through its derivatives with respect to the concentrations c j of its independent components, their temperature and pressure, then equation (14) can take the form: J k = – D k ( Σ j μ kj * c j + s k * T + υ k * p ). (15) where μ kj * ≡ ( ∂ μ k / ∂ c j ), s k * ≡ ( ∂ μ k / ∂ T ), υ k * ≡ ( ∂ μ k / ∂ р ). Three components of the resulting force F k on the right side of this expression handle the usual (concentration) diffusion F kc = Σ j μ kj * c j , thermal diffusion F kT = s k * T and baro diffusion F k р = υ k * p. This allows one to separate the thermodynamic factors μ kj, s k * , υ k * and the kinetic factors D k of multi component diffusion and establish a number of empirically established relationships between them[15]. Given the existing experimental means, it was mathematically unsound to 1 Year 2023 1 © 2023 Global Journals Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) New Applications of Non-Equilibrium Thermodynamics