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1 Title of communication titles not fitting in one line will break automatically First Author Second Author 2 Department University City Country 2 Other Institute City Country Abstract If you want to add some references insert single citations as [] or Author [2]; multiple citations as [2 3]. Text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract. Use off-text formulae if needed v = 0 () Reference them as in Eq.(). Text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract text of the abstract. Use empty lines to separate paragraphs. Introduction Entropy from the Greek ɛν (in) and τ ρoπη (turn transformation) is the state function related to the second principle of thermodyamics and was introduced by Clausius in early 850 for the historical bibliography see Wehrl [4] and references therein. As main thermodynamics properties we recall that: (i) The entropy variation is a measure of the amount of energy in the physical system that cannot be used to do useful work in a thermodynamic process; i.e. work mediated by thermal energy. (ii) An irreversible spontaneous evolution of a thermodynamic isolated system implies a net increase of its entropy. Edipo a Colono Edipo a Colono

2 flushleft. Later on scientists such as Ludwig Boltzmann Josiah Willard Gibbs and James Clerk Maxwell gave entropy a statistical basis. For the case of classical statistics in 877 Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of ideal gas particles in which he defined entropy to be proportional to the logarithm of the number of microstates such a gas could occupy. His celebrated equation gives for the entropy of an ensemble of ideal gas of particles: S = k B ln Γ (2) S = k B ln Γ S = k B ln Γ with k B the Boltzmann constant and Γ the number of microscopic states compatible with the macroscopic state of the system. ε ε 2 Σ σ A F Ã P 2 Σ σ A F Ã P f 2 f d f x i x i x k d x d 2 f d x 2 f 2 f d f x i x i x k d x d 2 f d x 2 With the development of quantum statistical mechanics a definition of entropy was formulated by von Neumann in 927 and is generally referred to as von Neumann entropy namely S = k B T r(ρ ln ρ) (3) with ρ the matrix density in Fock space T r the trace function with the constraint T rρ =.. The principle of entropy In Bayesian probability the principle of maximum entropy (MEP) is a postulate which states that subject to known constraints (called testable information) the probability distribution which best represents the current state of knowledge is the one with largest entropy. Let some testable information about a probability distribution function be given. Consider the set of all trial probability distributions that encode this information. Then the probability distribution that maximizes the information entropy is the true probability distribution with respect to the testable information prescribed. The principle was first expounded by Jaynes [?] where he emphasized a natural correspondence between statistical mechanics and information theory. 2

3 .. Gibbsian method of statistical mechanics. In particular Jaynes offered a new and very general rationale why the Gibbsian method of statistical mechanics works. He argued that the entropy of statistical mechanics and the information entropy of information theory are principally the same thing. As a consequence statistical mechanics should be seen just as a particular application of a general tool of logical inference and information theory. In most practical cases the testable information is given by a set of conserved quantities (average values of some moment functions) associated with the probability distribution in question. This is the way the MEP is most often used in statistical thermodynamics. probability distribution. Another possibility is to prescribe some symmetries of the probability distribution. An equivalence between the conserved quantities and corresponding symmetry groups implies the same level of equivalence for both these two ways of specifying the testable information in the maximum entropy method. The MEP is also needed to guarantee the uniqueness and consistency of probability assignments obtained by different methods statistical mechanics and logical inference in particular. probability density Strictly speaking the trial distributions which do not maximize the entropy are actually not probability distributions. The MEP makes explicit our freedom in using different forms of prior information. As a special case a uniform prior probability density (Laplace principle of indifference) may be adopted. Thus the MEP is not just an alternative to the methods of inference of classical statistics but it is an important conceptual generalization of those methods. 2 The maximum entropy principle In the last decades an accurate analysis of fluid transport in gas dynamics has been assuming an increasing importance in the description of many physical phenomena. Thus in the framework of a local theory various hydrodynamical models have been applied in many branches of physics for classic and quantum particles both in relativistic and not relativistic conditions (for example in the study of radiation transfer in relativistic fluids in the heat transport in dielectrics in the transport phenomena for the hot carriers in semiconductors in the analysis of thermodynamical properties of superfluids in the analysis of shock phenomena and so on). In particular if quantum methods are applied to molecular encounters then some divergence from the classical results appear and in some cases it is necessary to modify the classical theory in order to account for the quantum effects which are present in the collisional processes. In this section we first consider the Boltzmann transport equation (BTE) for a gas constituted by a 3

4 single specie of particles bosons or fermions. Consequently we propose a local theory based on the MEP and on the extended thermodynamics (ET) to describe the state of gas in which the moment of the distribution function are considered as basic fields. Usually the balance equations for numerical density velocity and energy density of the particles are used widely in literature by considering the moments of tensorial order 0 and 2 of the BTE. Analogously by taking into account moments up to the third order can be obtained the well-known 3 moments theory both in the framework of a linear (Grad theory) and nonlinear approach. Within the same strategy other authors have studied the behavior of classical and degenerate gases by adding also an increasing number of moments of higher order in the framework of local theory. 2. General local theory within a temperature scheme Extended thermodynamics provides a systematic method to obtain the constitutive equations present in the hierarchy of moments following two approaches which have proved to be equivalent the Entropy Principle (EP) and the MEP. f r (t u (t)) = f r (t) u r(t) 0 (4) f r (t) < f r (t u (t)) < f r (t) u r(t) = 0 (5) f r (t u (t)) = f r (t) u r(t) 0. (6) f r (t u (t)) = f r (t) u r(t) 0 f r (t) < f r (t u (t)) < f r (t) u r(t) = 0 (7) f r (t u (t)) = f r (t) u r(t) 0. f r (t u (t)) = f r (t) u r(t) 0 f r (t) < f r (t u (t)) < f r (t) u r(t) = 0 f r (t u (t)) = f r (t) u r(t) 0. The EP is based on the assumption that the entropy production of the system is nonnegative and consequently the entropy is an increasing function of time along the trajectories. It provides a powerful constraint to select the physical constitutive equations in the case of classical solutions and it becomes a fruitful selection rule for admissible weak solutions. Furthermore if the EP is combined with the stability requirement of the concavity of the entropy density then it permits to rewrite the field equations in the form of a symmetric hyperbolic system through the introduction of the privileged main field components. 4

5 The MEP allows one to derive the nonequilibrium distribution function associated with particles and to determine the microstate corresponding to the given macroscopic quantity. In this case the microscopic state is obtained from the solution of the variational problem of maximizing the entropy of the system under the constraints corresponding to the value of some mean quantities that define the macroscopic state. Once the distribution function is known all the unknown constitutive functions are obtained by integrating in momentum space their kinetic expression. We remak that the MEP can be exploited in the completely nonlinear case without any assumptions about the nonequilibrium processes. Θ ij k = 4 { } 8 5 S i δ j k + 5 W S i Σ jk 3 v i Σ jk + (8) { 27 m n W v i v j S k 3 m n v i v j v k + 3 [v r Σ 25 W rs (Σ si δ jk + Σ sk δ ij + Σ sj δ ki + 6 [v r (Σ 5 W ri Σ jk + Σ rk Σ ij + Σ rj Σ ki 3 ] [v i Σ 2 W jr Σ rk + v k Σ ir Σ rj + v j Σ kr Σ ri + 27 [ ( 25 W 2 S r Σ rs Σ si δ jk + Σ sk δ ij + Σ sj δ ki 36 [ ( 25 W 2 S r Σ ri Σ jk + Σ rk Σ ij + Σ rj Σ ki + 9 [ ] } 0 W 2 S i Σ jr Σ rk + S k Σ ir Σ rj + S j Σ kr Σ ri. Alternatively an approximate distribution function is usually derived through a formal expansion around to a local equilibrium configuration and so we obtain ET theories of N moments and degree α (ETN α theories). For example the Grad approach corresponds to N = 3 and α =. In the usual gas-dynamics the fully nonlinear closure suffers by some analytical problems that were firstly discovered by Junk and coworkers in what concerns the domain D of invertibility between the field variables and the Lagrange multipliers and the integrability of the moments. By contrast ETN α does not have this kind of problems since all the expressions for the moments are integrable. Accordingly from now on we will forget the original full nonlinear problem which justifies in a formal sense the expansion and consider explicitly only the ETN α theories in the case of degenerate Fermi and Bose gases. Finally by using the Galilean invariance principle it is possible to decompose both the basic fields and the constitutive functions in their convective 5

6 and nonconvective parts respectively. Γ αβ = Γ (0) w A (0) B (0) Γ () w C () A () B () Γ (2) 0 C (2) A (2) B (2) Γ (3) w 0 0 C (3) A (3) B (3) 0 0 Γ (4) w C (4) A (4) B (4) 0 Γ (5) w C (5) A (5) 0 Table : Device parameters with parabolic band structure Dev. N + (cm 3 ) N (cm 3 ) Channel (µm) bias (V) A B C D E This decomposition has the advantage of allowing the introduction of a state equation and in turn the local temperature concept for the gas. In this case it is possible to define the usual local equilibrium Fermi and/or Bose distribution function in terms of numerical density n(r t) and of temperature T (r t). Analogously for all the constitutive functions the dependence on the moments of the distribution function is of local-type and the evolution equations for suitable values of the independent variables determine a quasilinear hyperbolic system within a temperature scheme. A Being w (0) (ξ) solution of (3) (with ξ = e α+x2 ) we define the integrals I n (α κ) = + x n [w (0) (e α+x2 ) + κ] dx where for n < 0 all the integral functions 0 I n (α κ) can be obtained by means of the following general differentiation property r I n / α r = ( ) [ r Γ ( ) ( n+ 2 /Γ n+ 2 r I n 2r. The functions η (s) ij are given by B ij = ( ) i 2 j Γ ( D 2 + s + j ) Γ ( I D+2(s+i+j) D 2 + s + i + j 5) I D+2s η (s) 6

7 C The quantum gradient corrections terms in are [ ( ) ] 2 Q II D = h2 2D 2 ln n ln n + (D 4) 2 D m x r x r x r Q II E + h2 32 = h2 2D 2 Tll D(D 6) [ m M II ik = h2 n 2 m 2 µ (0) 2 ln n (D 3) + x r x r D [ 4 ln n ln n + m D x i x j 2 ( ) ] 2 ln n h2 x r 32 ] D µ T 2 (0) ik T 2 ll µ (0) References [] Bookauthor A. B.: Book title book title book title. Book publisher Publisher address (Year) [2] Firstauthor J. K. Secondauthor B. Thirdauthor C.: Article title Article title Article title. Rev. Mat. 99 n (Year) [3] Author D.: Article title Article title Article title. Appl. Mat. Rev (Year) [4] Wehrl A. Rev. Mod. Phys (978). 7