The Euler limit for kinetic models with Fermi

Abstract

We are interested in the connection between kinetic models with Fermi–Dirac statistics and fluid dynamics. We establish that moments and parameters of Fermi–Dirac distributions are related by a diffeomorphism. We obtain the macroscopic limits when the fluid is dense enough that particles undergo many collisions per unit of time. This situation is described via a small parameter ε, called the Knudsen number, that represents the ratio of mean free path of particles between collisions to some characteristic length of the flow. We give the conditions that allow us to formally derive the generalized Euler equations from the Boltzmann equation by adopting the formalism proposed in [Advances in Kinetic Theory and Continuum Mechanics, Springer, Berlin, 1991, pp. 57–71]. These conditions are related to the H-theorem and assume a formally consistent convergence for fluid dynamical moments and entropy of the kinetic equation. We also discuss the well-posedness of the obtained Euler equations by using Godunov’s criterion of hyperbolicity.

Keywords

kinetic model Boltzmann equation Euler equations entropy

1. Introduction

In this work we establish the connection between kinetic theory for Fermi–Dirac statistics and macroscopic fluid dynamics. We derive formal limits; in order to do that, we introduce a scaling for standard kinetic equation (see, for example, [12]) of the form $\partial_{t} F_{ε} + v \cdot \nabla_{x} F_{ε} = \frac{1}{ε} C (F_{ε}) .$ (1.1) Here $F_{ε}$ is a non-negative function representing the density of particles with position x and velocity v in the single-particle phase space $R_{x}^{3} \times R_{v}^{3}$ at time t. The interaction of particles through collisions is given by the operator $C (F)$ ; this operator acts only on variable v and is non-linear in the general case. We will keep this operator abstract.

We base the connection between kinetic and macroscopic dynamics on the conservation properties and entropy relations implying that the equilibria are Fermi–Dirac (i.e. of the form $1 / (1 + exp (c_{1} + c_{2} | v - u |^{2}))$ ) distributions.

It is important to notice that our approach differs from Hilbert expansion employed in works of C. Cercignani. Our results highlight the role of entropy as it was done in [1]; the convergence assumptions are similar to those in [3]. We also adopt the formalism for moments of distributions proposed in [2].

In the Section 4 we examine the moments of Fermi–Dirac distributions and establish that the parameters of such a distribution are related to the moments by a diffeomorphism. While in the case of Maxwellian distributions such a relation is rather evident, the case of Fermi–Dirac distributions requires an additional analysis.

We obtain the macroscopic limits when the fluid is dense enough that particles undergo many collisions per unit of time. In order to describe this situation, we introduce a small parameter ε, called the Knudsen number, that represents the ratio of mean free path of particles between collisions to some characteristic length of the flow.

Conservation properties are used to derive the compressible Euler equations from (1.1); we will do so in Section 6, assuming a formally consistent convergence for fluid dynamical moments and entropy of the kinetic Eq. (1.1) (see Theorem 5).

2. Kinetic models with Fermi–Dirac statistics

Denote for an integrable function s its moment $⟨ s (v) ⟩ = \int_{R^{3}} s (v) d v$ .

We assume that for all measurable functions rapidly decaying on infinity $F (t, x, v) : R_{+} \times R^{3} \times R^{3} \to R, 0 ⩽ F ⩽ 1 a.e.$ (2.1) the collision operator C satisfies the conservation properties $⟨ C (F) ⟩ = 0, ⟨ C (F) v ⟩ = 0, ⟨ C (F) | v |^{2} ⟩ = 0,$ (2.2) corresponding to the conservation of mass, momentum and energy through the collision process. If we multiply Eq. (1.1) by 1, v, $| v |^{2}$ and integrate with respect to v, we obtain the respective local conservation laws: $\begin{array}{l} \partial_{t} ⟨ F_{ε} ⟩ + \nabla_{x} \cdot ⟨ F_{ε} v ⟩ = 0, \\ \partial_{t} ⟨ F_{ε} v ⟩ + \nabla_{x} \cdot ⟨ F_{ε} v \otimes v ⟩ = 0, & (2.3) \\ \partial_{t} ⟨ F_{ε} | v |^{2} ⟩ + \nabla_{x} \cdot ⟨ F_{ε} | v |^{2} v ⟩ = 0 . \end{array}$

We also assume that for every measurable rapidly decaying function F satisfying (2.1) the non-negative quantity $⟨ C (F) ln (\frac{1 - F}{F}) ⟩ ⩾ 0$ (2.4) is the entropy production rate for this collision process.

Observe that $\begin{array}{rcl} \partial_{t} (F_{ε} ln F_{ε} + (1 - F_{ε}) ln (1 - F_{ε})) = (ln F_{ε} - ln (1 - F_{ε})) \partial_{t} F_{ε}, \\ \nabla_{x} (F_{ε} ln F_{ε} + (1 - F_{ε}) ln (1 - F_{ε})) = (ln F_{ε} - ln (1 - F_{ε})) \nabla_{x} F_{ε}, \end{array}$ therefore multiplying both parts of Eq. (1.1) by $ln (\frac{F_{ε}}{1 - F_{ε}})$ and integrating with respect to v yields $\begin{array}{rcl} ⟨ ln (\frac{1 - F_{ε}}{F_{ε}}) \partial_{t} F_{ε} + ln (\frac{1 - F_{ε}}{F_{ε}}) v \cdot \nabla_{x} F_{ε} ⟩ \\ = ⟨ \partial_{t} (F_{ε} ln F_{ε} + (1 - F_{ε}) ln (1 - F_{ε})) + v \cdot \nabla_{x} (F_{ε} ln F_{ε} + (1 - F_{ε}) ln (1 - F_{ε})) ⟩ \\ = - ⟨ C (F_{ε}) ln (\frac{1 - F_{ε}}{F_{ε}}) ⟩ ⩽ 0, \end{array}$ which gives us a local entropy inequality $\partial_{t} ⟨ (F_{ε} ln F_{ε} + (1 - F_{ε}) ln (1 - F_{ε})) ⟩ + \nabla_{x} \cdot ⟨ (v F_{ε} ln F_{ε} + v (1 - F_{ε}) ln (1 - F_{ε})) ⟩ ⩽ 0 .$

Finally, the equilibria are assumed to be characterized by zero entropy production rate and are given by the class of Fermi–Dirac distributions (for more information see, for example, [11]) $F (v) = {(1 + exp (\frac{| v - u |^{2}}{2 θ} - f))}^{- 1}, u \in R^{3}, f \in R, θ > 0 .$ (2.5) We assume the following analogy of the Boltzmann’s H-theorem for the Fermi–Dirac statistics. Theorem 1.

For every measurable rapidly decaying function F satisfying (2.1) with at most polynomially increasing $| ln (\frac{1 - F}{F}) |$ the following properties are equivalent: $\begin{matrix} (1) & C (F) = 0, \\ (2) & ⟨ C (F) ln (\frac{1 - F}{F}) ⟩ = 0, \\ (3) & F is a Fermi–Dirac distribution of the form (2.5) . \end{matrix}$ (2.6)

3. Example of a collision operator for Fermi–Dirac statistics

In this section we give an example of a collision operator satisfying the conservation properties (2.2) and with positive entropy production rate (2.4) together with H-theorem.

Consider the operator studied in [5]: $C (F) = \iint_{R^{3} \times S^{2}} b (F^{'} F_{*}^{'} (1 - F) (1 - F_{*}) - F F_{*} (1 - F^{'}) (1 - F_{*}^{'})) d v_{*} d ω,$ (3.1) where for $ω \in S^{2}$ $\begin{array}{rcl} v^{'} = v - ω (v - v_{*}, ω), v_{*}^{'} = v_{*} + ω (v - v_{*}, ω), \\ F^{'} = F (t, x, v^{'}), F_{*} = F (t, x, v_{*}), F_{*}^{'} = F (t, x, v_{*}^{'}) \end{array}$ and $b = b (| v - v_{*} |, | (ω, v - v_{*}) |)$ – a collision kernel. Observe that the definition of vectors $v^{'}$ and $v_{*}^{'}$ implies the following relations: $v^{'} + v_{*}^{'} = v + v_{*}, {| v^{'} |}^{2} + {| v_{*}^{'} |}^{2} = | v |^{2} + | v_{*} |^{2} .$ (3.2) Observe also that $| v - v_{*} | = | v^{'} - v_{*}^{'} |, | (v - v_{*}, ω) | = | (v^{'} - v_{*}^{'}, ω) | .$ The above expression allows us to take the collision kernel in the form ${\bar{b}}_{ω} (v, v_{*}, v^{'}, v_{*}^{'}) = b (\frac{1}{2} | v - v_{*} | + \frac{1}{2} | v^{'} - v_{*}^{'} |, \frac{1}{2} | (ω, v - v_{*}) | + \frac{1}{2} | (v^{'} - v_{*}^{'}, ω) |) .$ (3.3) In order to prove that the operator C indeed satisfies the relations (2.2) and (2.4), we first establish the following proposition.

Proposition 1.

Suppose that we have functions $W : {(R^{3})}^{4} \to C$ and $h : R^{3} \to C$ such that the integral $I = ∭_{R^{3} \times R^{3} \times S^{2}} W (v, v_{*}, v - (v - v_{*}, ω) ω, v_{*} + (v - v_{*}, ω) ω) h (v) d v d v_{*} d ω$ exists. Then the following identity holds: $\begin{array}{rcl} I & = & ∭_{R^{3} \times R^{3} \times S^{2}} W (v_{*}, v, v_{*} + (v - v_{*}, ω) ω, v - (v - v_{*}, ω) ω) h (v_{*}) d v d v_{*} d ω \\ = & ∭_{R^{3} \times R^{3} \times S^{2}} W (v^{'}, v_{*}^{'}, v, v_{*}) h (v^{'}) d v d v_{*} d ω \\ = & ∭_{R^{3} \times R^{3} \times S^{2}} W (v_{*}^{'}, v^{'}, v_{*}, v) h (v_{*}^{'}) d v d v_{*} d ω . \end{array}$

We give the proof of this proposition in Appendix B.

The following proposition shows that the operator C indeed satisfies the conservation properties (2.2). Proposition 2.

$⟨ C (F) ⟩ = 0, ⟨ v C (F) ⟩ = 0, ⟨ | v |^{2} C (F) ⟩ = 0$ for any measurable function F rapidly decaying on infinity satisfying (2.1) .

Proof.

Take any function $h : R^{3} \to R$ , $h (v) \in span {1, v_{1}, v_{2}, v_{3}, | v |^{2}}$ . Let $w_{i} \in R^{3}$ ; introduce a function $\begin{array}{l} W (w_{1}, w_{2}, w_{3}, w_{4}) = & \bar{b} (w_{1}, w_{2}, w_{3}, w_{4}) F (w_{3}) F (w_{4}) (1 - F (w_{1})) (1 - F (w_{2})) \\ - \bar{b} (w_{1}, w_{2}, w_{3}, w_{4}) F (w_{1}) F (w_{2}) (1 - F (w_{3})) (1 - F (w_{4})) & (3.4) \end{array}$ and observe that $\begin{matrix} W (w_{1}, w_{2}, w_{3}, w_{4}) & = W (w_{2}, w_{1}, w_{3}, w_{4}) \\ = W (w_{1}, w_{2}, w_{4}, w_{3}) = - W (w_{3}, w_{4}, w_{1}, w_{2}) \end{matrix}$ for all $w_{i}$ . Applying Proposition 1 to W defined above and h we can conclude that $\begin{matrix} ⟨ h (v) C (F) ⟩ = & - \frac{1}{4} ∭_{R^{3} \times R^{3} \times S^{2}} b (F^{'} F_{*}^{'} (1 - F) (1 - F_{*}) - F F_{*} (1 - F^{'}) (1 - F_{*}^{'})) \\ \cdot (h (v^{'}) + h (v_{*}^{'}) - h (v) - h (v_{*})) d v d v_{*} d ω . \end{matrix}$ Since $h (v) \in span {1, v_{1}, v_{2}, v_{3}, | v |^{2}}$ , applying the relation (3.2) yields $h (v^{'}) + h (v_{*}^{'}) - h (v) - h (v_{*}) = 0$ for all v, $v_{*}$ and ω. Therefore $⟨ C (F) h (v) ⟩ = 0$ and the proposition holds. □

Proposition 3.

For any measurable function F rapidly decaying on infinity satisfying (2.1) we have $⟨ C (F) ln (\frac{1 - F}{F}) ⟩ ⩾ 0 .$

Proof.

We apply the Proposition 1 with W defined in (3.4) and $h = ln (\frac{1 - F}{F})$ and obtain that $⟨ C (F) ln (\frac{1 - F}{F}) ⟩ = \frac{1}{4} ⟨ C (F) ln (\frac{1 - F}{F} \frac{1 - F_{*}}{F_{*}} \frac{F^{'}}{1 - F^{'}} \frac{F_{*}^{'}}{1 - F_{*}^{'}}) ⟩ .$ Denote $X = (1 - F) (1 - F_{*}) F^{'} F_{*}^{'}$ and $Y = F F_{*} (1 - F^{'}) (1 - F_{*}^{'})$ , then we can rewrite the above expression as $\frac{1}{4} ∭_{R^{3} \times R^{3} \times S^{2}} b (X - Y) ln (X / Y) d v d v_{*} d ω .$ The function $(X, Y) \to (X - Y) ln (X / Y)$ is non-negative and so is the collision kernel b, therefore we can conclude that $⟨ C (F) ln (\frac{1 - F}{F}) ⟩ ⩾ 0 .$ □

Proposition 4 (H-theorem).

For every measurable rapidly decaying F with at most polynomially increasing $| ln (\frac{1 - F}{F}) |$ satisfying (2.1) the following properties are equivalent: $\begin{matrix} (1) & C (F) = 0, \\ (2) & ⟨ C (F) ln (\frac{1 - F}{F}) ⟩ = 0, \\ (3) & F is a Fermi–Dirac distribution of the form (2.5) . \end{matrix}$ (3.5)

Proof.

If F is of the form (2.5), then $ln (\frac{1 - F}{F}) \in span {1, v_{1}, v_{2}, v_{3}, | v^{2} |},$ hence the entropy production rate is zero. On the other hand, by direct substitution one can show that in this case $C (F) = 0$ .

If the entropy production rate is zero, then applying the Proposition 1 with W defined in (3.4) and $h = ln (\frac{1 - F}{F})$ yields $⟨ C (F) ln (\frac{1 - F}{F} \frac{1 - F_{*}}{F_{*}} \frac{F^{'}}{1 - F^{'}} \frac{F_{*}^{'}}{1 - F_{*}^{'}}) ⟩ = 0 .$ Let for simplicity $G = \frac{F}{1 - F}$ , then we can rewrite this expression as $∭_{R^{3} \times R^{3} \times S^{2}} b (1 - F) (1 - F_{*}) (1 - F^{'}) (1 - F_{*}^{'}) \cdot (G_{*}^{'} G^{'} - G_{*} G) ln (\frac{G_{*}^{'} G^{'}}{G_{*} G}) d v d v_{*} d ω .$ Since the function $(X, Y) \to (X - Y) ln (X / Y)$ is non-negative on $R^{2}$ and vanishes if and only if $X = Y$ , we deduce that $G_{*}^{'} G^{'} = G_{*} G$ almost everywhere in v, or, in other words $ln G_{*}^{'} + ln G^{'} = ln G_{*} + ln G a.e.$ By Boltzmann–Gronwall theorem (see, for example, [4,6,16]) this implies that $ln G (v) = a + w \cdot v + c | v |^{2}$ for some constants $a, c \in R$ and $w \in R^{3}$ . In its turn, this implies that G is a local Maxwellian distribution and therefore F is a local Fermi–Dirac distribution.

Finally, if $C (F) = 0$ , $0 = C (F) = \iint_{R^{3} \times S^{2}} b (1 - F) (1 - F_{*}) (1 - F^{'}) (1 - F_{*}^{'}) \cdot (G_{*}^{'} G^{'} - G_{*} G) d v_{*} d ω$ (3.6) with $G = \frac{F}{1 - F}$ . Since the factor $b (1 - F) (1 - F_{*}) (1 - F^{'}) (1 - F_{*}^{'})$ is non-negative, we obtain that $G_{*}^{'} G^{'} - G_{*} G = 0$ , which, as above, implies that G is a local Maxwellian distribution, and therefore F is a Fermi–Dirac distribution. □

4. Moments and parameters of Fermi–Dirac distributions

Recall the definition of Maxwellian distributions: $M (v) = M_{ρ^{M}, u^{M}, θ^{M}} (v) = \frac{ρ^{M}}{{(2 π θ^{M})}^{3 / 2}} e^{- | v - u^{M} |^{2} / (2 θ^{M})}$ for parameters $ρ^{M} ⩾ 0$ , $u^{M} \in R^{3}$ , $θ^{M} > 0$ . Define the vector of moments for such a distribution: $(\begin{matrix} μ_{0} \\ μ_{1} \\ μ_{2} \end{matrix}) = (\begin{matrix} {⟨ 1 ⟩}_{M} \\ {⟨ v ⟩}_{M} \\ {⟨ | v |^{2} ⟩}_{M} \end{matrix}) = (\begin{matrix} ρ^{M} \\ ρ^{M} u^{M} \\ ρ^{M} | u^{M} |^{2} + 3 ρ^{M} θ^{M} \end{matrix}) .$ (4.1) It is easy to see that the space of possible moments is described by $μ_{0} ⩾ 0, μ_{1} \in R^{3}, μ_{0} μ_{2} ⩾ | μ_{1} |^{2} .$ Moreover, the map $T_{M} : R^{5} \to R^{5}, T_{M} (ρ^{M}, u^{M}, θ^{M}) = (μ_{0}, μ_{1}, μ_{2})$ is a diffeomorphism; indeed, the expression (4.1) clearly shows that $T_{M}$ is smooth. In addition, the same expression allows us to say that $\begin{array}{l} ρ^{M} = μ_{0}, \\ u^{M} = \frac{μ_{1}}{μ_{0}}, \\ θ^{M} = \frac{μ_{2} - | μ_{1} |^{2} / μ_{0}}{3 μ_{0}}, \end{array}$ hence the inverse application exists and is also smooth.

Another observation requires a slightly different parametrisation of a Maxwellian distribution. If $M (v) = e^{- | v - u^{M} |^{2} / (2 θ^{M}) - f^{M}},$ and if the moments are defined as $ρ^{M} = ⟨ M (v) ⟩, E^{M} = \frac{1}{3} ⟨ | v - u |^{2} M (v) ⟩,$ then the following equations hold: $\begin{array}{rcl} ρ^{M} \partial_{ρ^{M}} f^{M} + \frac{5}{3} E^{M} \partial_{E^{M}} f^{M} = 0, \\ ρ^{M} \partial_{ρ^{M}} θ^{M} + \frac{5}{3} E^{M} \partial_{E^{M}} θ^{M} = \frac{2}{3} θ^{M} . \end{array}$ This result quickly follows from the relations $\begin{array}{rcl} f^{M} = - ln (\frac{ρ^{M}}{{(2 π θ^{M})}^{3 / 2}}), \\ ρ^{M} = μ_{0}, E^{M} = \frac{μ_{2} - | μ_{1} |^{2} / μ_{0}}{3} . \end{array}$

The goal of this section is to establish the similar properties of moments of Fermi–Dirac distributions and find under which conditions the parameters of this distribution can be found via its moments.

We study the Fermi–Dirac distribution $F_{f, u, θ} (v) = \frac{1}{1 + exp (| v - u |^{2} / (2 θ) - f)} .$ The admissible parameters form an open convex set $D \subset R^{5}$ : $(f, u, θ) \in D = R \times R^{3} \times (0, \infty) .$ The moments of the distribution F are $⟨ F (v) ⟩ ⩾ 0, ⟨ v F (v) ⟩ \in R^{3}, ⟨ | v |^{2} F (v) ⟩ ⩾ 0 .$ Therefore, we can define a map $T_{F} : D \to R^{5}, T (f, u, θ) = (⟨ F (v) ⟩, ⟨ v F (v) ⟩, ⟨ | v |^{2} F (v) ⟩) .$

Let U be the range of $T_{F}$ , i.e. $U = T_{F} (D)$ . We will examine the properties of U and establish that $T_{F}$ is a diffeomorphism $D \to U$ . To simplify the calculations, we will introduce the following notations: $\begin{array}{rclr} ρ = \int_{R^{3}} \frac{d v}{1 + exp (| v - u |^{2} / (2 θ) - f)} = θ^{3 / 2} \int_{R^{3}} \frac{d v}{1 + exp (| v |^{2} / 2 - f)}, & (4.2) \\ E = \frac{1}{3} \int_{R^{3}} \frac{| v - u |^{2} d v}{1 + exp (| v - u |^{2} / (2 θ) - f)} = \frac{θ^{5 / 2}}{3} \int_{R^{3}} \frac{| v |^{2} d v}{1 + exp (| v |^{2} / 2 - f)} . & (4.3) \end{array}$ It is easy to see that $T_{F} (f, u, θ) = (ρ, ρ u, ρ | u |^{2} + 3 E) .$

The quantities ρ and $E$ can be expressed in terms of polylogarithms assuming $⟨ F (v) ⟩ \neq 0$ . We discuss the properties of these special functions in Appendix A.

The following theorem establishes the invertibility of the map $T_{F}$ .

Theorem 2.

If ρ and $E$ are given by (4.2) and (4.3), respectively, then the following statements hold:

the ratio $\frac{ρ}{E^{3 / 5}}$ depends only on f,

setting $J = \frac{{(8 π \sqrt{2})}^{2 / 5}}{3} {(\frac{5}{2})}^{3 / 5}$ , then the map $f \to \frac{ρ}{E^{3 / 5}}$ is $C^{\infty} (R; (0, J))$ and strictly monotone; there exists a function $\bar{f} \in C^{1} ((0, J), R), f = \bar{f} (\frac{ρ (f, θ)}{{(E (f, θ))}^{3 / 5}}),$ (4.4)

the function θ is given by $θ = {(\frac{ρ}{4 \sqrt{2} π Γ (3 / 2) F_{3 / 2} (f)})}^{2 / 3}$ ,

the map $T_{F}$ is a diffeomorphism,

f and θ seen as functions of ρ and $E$ satisfy $\begin{array}{rclr} ρ \partial_{ρ} f + \frac{5}{3} E \partial_{E} f = 0, & (4.5) \\ ρ \partial_{ρ} θ + \frac{5}{3} E \partial_{E} θ = \frac{2}{3} θ . & (4.6) \end{array}$

Proof.

We can express ρ and $E$ in terms of polylogarithms: $\begin{array}{rclr} ρ (f, θ) = 4 π \sqrt{2} θ^{3 / 2} Γ (3 / 2) F_{3 / 2} (f), & (4.7) \\ E (f, θ) = 8 π \sqrt{2} θ^{5 / 2} Γ (5 / 2) F_{5 / 2} (f), & (4.8) \end{array}$ with $F_{p} (w) = \frac{1}{Γ (p)} \int_{0}^{\infty} \frac{t^{p - 1}}{e^{t - w} + 1} d t, G_{p} (w) = {(F_{p} (w))}^{1 / p},$ which yields (1). Let $a = \frac{ρ}{E^{3 / 5}} .$ The expressions for moments together with properties of polylogarithms yield $a = \frac{4 π \sqrt{2} Γ (3 / 2)}{{(8 π \sqrt{2} Γ (5 / 2))}^{3 / 5}} \frac{F_{3 / 2} (f)}{{(F_{5 / 2} (f))}^{3 / 5}} = \frac{5}{2} \frac{4 π \sqrt{2} Γ (3 / 2)}{{(8 π \sqrt{2} Γ (5 / 2))}^{3 / 5}} \frac{d}{d f} G_{5 / 2} (f) .$

As established in the Theorem A.1 (see Appendix A), the function $f \to \frac{d}{d f} G_{5 / 2} (f)$ is strictly monotone, hence the map $f \mapsto a$ is invertible. In order to obtain possible values of a, we study $lim_{f \to \pm \infty} \frac{F_{3 / 2} (f)}{{(F_{5 / 2} (f))}^{3 / 5}} .$ The Taylor development of polylogarithms for $| z | < 1$ writes (see, for example, [13–15]) $- {Li}_{s} (- z) = \frac{1}{Γ (s)} \int_{0}^{\infty} \frac{t^{s - 1} d t}{e^{t} / z + 1} = \sum_{k ⩾ 1} \frac{z^{k}}{k^{s}},$ in other words, $- {Li}_{s} (- z) = z + O (z^{2})$ near $z = 0$ , which implies that $lim_{f \to - \infty} \frac{F_{3 / 2} (f)}{{(F_{5 / 2} (f))}^{3 / 5}} = 0 .$

On the other hand, we know the limiting behaviour of polylogarithms when $f \to + \infty$ and $p \neq - 1, - 2, - 3, \dots$ (see, for example, [17]): $lim_{f \to + \infty} \frac{F_{p} (f)}{f^{p}} = \frac{1}{Γ (p + 1)},$ which yields $lim_{f \to \infty} \frac{F_{3 / 2} (f)}{{(F_{5 / 2} (f))}^{3 / 5}} = \frac{{(Γ (7 / 2))}^{3 / 5}}{Γ (5 / 2)},$ hence $a < \frac{4 π \sqrt{2} Γ (3 / 2)}{{(8 π \sqrt{2} Γ (5 / 2))}^{3 / 5}} \frac{{(Γ (7 / 2))}^{3 / 5}}{Γ (5 / 2)} = \frac{{(8 π \sqrt{2})}^{2 / 5}}{3} {(\frac{5}{2})}^{3 / 5} = J .$ Thanks to the monotonicity of the function $f \to \frac{d}{d f} G_{5 / 2} (f)$ , we obtain that $a \in (0, J)$ . The regularity of the function $\bar{f}$ follows from the regularity of polylogarithms; the expression for θ is a direct consequence of (4.7) and the previous point.

Finally, the following equations show that the map $T_{F}$ is invertible: $\begin{array}{rcl} u = \frac{⟨ v F (v) ⟩}{ρ}, \\ E = \frac{1}{3} (⟨ | v |^{2} F (v) - ρ | u |^{2} ⟩), \\ f = \bar{f} (\frac{ρ}{E^{3 / 5}}), \\ θ = {(\frac{ρ}{4 \sqrt{2} π Γ (3 / 2) F_{3 / 2} (f)})}^{2 / 3} . \end{array}$

The differential Eqs (4.5) and (4.6) follow from (1) and (2), respectively. □

Remark.

The inequality $\frac{ρ}{E^{3 / 5}} < J = {(\frac{5}{2})}^{3 / 5} {(4 π \frac{2 \sqrt{2}}{3})}^{2 / 5}$ (4.9) comes from the form of Fermi–Dirac distributions; they are bounded by 1, therefore this inequality can be interpreted as that a Fermi–Dirac distribution cannot accumulate an arbitrary number of particles with small velocities in one point.

The result of the Theorem 2 allows us to say that the image of the map $T_{F}$ is a convex set $U = {(x, v, z) \in (0, \infty) \times R^{3} \times R : z > \frac{| v |^{2}}{x} + J^{5 / 3} x^{5 / 3}} .$ (4.10) Indeed, U is the subset of $R^{5}$ above the graph of the function $h : (0, \infty) \times R^{3} \to R, h (x, v) = \frac{| v |^{2}}{x} + J^{5 / 3} x^{5 / 3} .$ The Hessian of h writes $Hess (h) = \frac{2}{x^{2}} (\begin{matrix} | v |^{2} + \frac{5}{9} J^{5 / 3} x^{5 / 3} & - v_{1} & - v_{2} & - v_{3} \\ - v_{1} & x & 0 & 0 \\ - v_{2} & 0 & x & 0 \\ - v_{3} & 0 & 0 & x \end{matrix}) .$ By Sylvester’s criterion, this matrix is positive definite for $(x, v) \in (0, \infty) \times R^{3}$ ; hence, the function h is convex and, therefore, the set U is convex.

5. The entropy and the map

T_{F}

From now on we drop the subscript in the notation $T_{F}$ . We define the vector of conserved quantities $\vec{e}$ as follows: $\vec{e} (v) = (\begin{matrix} 1 \\ v_{1} \\ v_{2} \\ v_{3} \\ | v |^{2} \end{matrix}) .$ Let us define the operator ⊙ as the standard scalar product in $R^{5}$ . Then we can put every Fermi–Dirac distribution in the form $F [\vec{β}] = {(1 + e^{\vec{β} ⊙ \vec{e}})}^{- 1}, \vec{β} \in D = R \times R^{3} \times (0, \infty) .$ It is important to notice that $\vec{β}$ does not depend on v.

One can easily see that the parameters $(f, u, θ)$ of the representation (2.5) and $\vec{β}$ are connected by $\begin{matrix} β_{4} = \frac{1}{2 θ}, & θ = \frac{1}{2 β_{4}}, \\ β_{3} = - \frac{u_{3}}{θ}, & u_{3} = - \frac{β_{3}}{2 β_{4}}, \\ β_{2} = - \frac{u_{2}}{θ}, & u_{2} = - \frac{β_{2}}{2 β_{4}}, \\ β_{1} = - \frac{u_{1}}{θ}, & u_{1} = - \frac{β_{1}}{2 β_{4}}, \\ β_{0} = - f + \frac{| u |^{2}}{2 θ}, & f = - β_{0} + (β_{1}^{2} + β_{2}^{2} + β_{3}^{2}) \frac{1}{4 β_{4}} . \end{matrix}$ (5.1) Let us consider the vector of conserved densities $\vec{ρ} = - ⟨ \vec{e} F [\vec{β}] ⟩ .$ (5.2) Clearly, the image of the map $T : \vec{β} \to \vec{ρ}$ is the set $U^{'} = - U$ where U is defined in (4.10). The relations (5.1) and Theorem 2 imply that the map T is a diffeomorphism $D \to U^{'}$ .

The following proposition establishes the form of entropy associated with the Fermi–Dirac distribution.

Theorem 3.

There exists a positive and strictly convex function $σ^{*} : D \to R$ such that $\vec{ρ} (\vec{β}) = \nabla_{\vec{β}} σ^{*} (\vec{β})$ for $\vec{ρ}$ given by (5.2) .

There exists a strictly convex function $σ : U^{'} \to R$ such that $\vec{β} (\vec{ρ}) = \nabla_{\vec{ρ}} σ (\vec{ρ});$ moreover, this function is the entropy density of the Fermi–Dirac distribution associated with $T^{- 1} (\vec{ρ})$ .

Proof.

Put $σ^{*} (\vec{β}) = - ⟨ ln (1 - F [\vec{β}]) ⟩,$ then $\begin{array}{rcl} \nabla_{\vec{β}} σ^{*} (\vec{β}) & = & - \nabla_{\vec{β}} ⟨ ln (1 - F [\vec{β}]) ⟩ = ⟨ \frac{\nabla_{\vec{β}} F [\vec{β}]}{1 - F [\vec{β}]} ⟩ \\ = & - ⟨ \frac{F [\vec{β}] (1 - F [\vec{β}]) \vec{e}}{1 - F [\vec{β}]} ⟩ = \vec{ρ} . \end{array}$ Note that $σ^{*}$ is a strictly positive and convex function of $\vec{β}$ , because $\nabla_{\vec{β}} \vec{ρ} = ⟨ \vec{e} \otimes \vec{e} F [\vec{β}] (1 - F [\vec{β}]) ⟩$ is a positive definite matrix. Indeed, take a vector $\vec{w} \in R^{5}$ and suppose that $0 = ⟨ \vec{e} \otimes \vec{e} F (\vec{β}) (1 - F [\vec{β}]) ⟩ \vec{w} ⊙ \vec{w} = ⟨ | \vec{e} ⊙ \vec{w} |^{2} F [\vec{β}] (1 - F [\vec{β}]) ⟩ .$ Since $F [\vec{β}] (1 - F [\vec{β}])$ is positive, then $\vec{e} ⊙ \vec{w} = 0$ for almost all v; the functions 1, $v_{1}$ , $v_{2}$ , $v_{3}$ and $| v |^{2}$ are linearly independent on $R^{3}$ , which results in $\vec{w} = 0$ . We can therefore conclude that $σ^{*}$ is strictly convex.

Let the function σ be the Legendre transform of the function $σ^{*}$ , then σ is also a convex function and $\vec{β} = \nabla_{\vec{ρ}} σ (\vec{ρ}) .$ The function σ is defined on $U^{'}$ and can be expressed via the relation $σ (\vec{ρ}) + σ^{*} (\vec{β}) = \vec{β} ⊙ \vec{ρ}$ with $\vec{β} = {(\nabla_{\vec{β}} σ^{*})}^{- 1} (\vec{ρ}) = T^{- 1} (\vec{ρ})$ , so $\begin{array}{rclr} σ (\vec{ρ}) & = & ⟨ - \vec{β} ⊙ \vec{e} F [\vec{β}] + ln (1 - F [\vec{β}]) ⟩ \\ = & ⟨ (1 - F [\vec{β}]) ln (1 - F [\vec{β}]) + F [\vec{β}] ln (F [\vec{β}]) ⟩ . & (5.3) \end{array}$ In other words, this function coincides with the expression for the entropy density of the Fermi–Dirac distribution associated with $T^{- 1} (\vec{ρ})$ . □

The following theorem establishes the form of the entropy flux associated with entropy density σ defined in Theorem 3. Theorem 4.

The flux corresponding to the Fermi–Dirac distribution $F [\vec{β}]$ for each $\vec{ρ} \in U^{'}$ , $\vec{β} = T^{- 1} (\vec{ρ})$ is a map $\vec{υ} : U^{'} \to R^{5 \times 3}$ , written as $\vec{υ} (\vec{ρ}) = - ⟨ \vec{e} \otimes v F [T^{- 1} (\vec{ρ})] ⟩ .$ There exists a function $τ^{*} : D \to R^{3}$ such that $\vec{υ} (T (\vec{β})) = {(\nabla_{\vec{β}} τ^{*} (\vec{β}))}^{T} .$

The function $τ : U^{'} \to R^{3}$ for every $\vec{ρ} \in U^{'}$ and $\vec{β} = T^{- 1} (\vec{ρ})$ defined by $τ (\vec{ρ}) = ⟨ v (1 - F [\vec{β}]) ln (1 - F [\vec{β}]) + v F [\vec{β}] ln (F [\vec{β}]) ⟩,$ $τ (\vec{ρ})$ is the entropy flux for the Fermi–Dirac distribution associated with conserved densities $\vec{ρ}$ .

Proof.

The expression for $\vec{υ} (\vec{ρ})$ quickly follows from the form of local conservation laws (2.3).

If $τ^{*} : D \to R^{3}, τ^{*} (\vec{β}) = - ⟨ v ln (1 - F [\vec{β}]) ⟩,$ then $\vec{υ} (\vec{ρ}) = {(\nabla_{\vec{β}} τ^{*} (\vec{β}))}^{T},$ with $\vec{β} = T^{- 1} (\vec{ρ})$ . Indeed, $\begin{array}{rcl} \nabla_{\vec{β}} τ^{*} (\vec{β}) & = & - \nabla_{\vec{β}} ⟨ v ln (1 - F [\vec{β}]) ⟩ \\ = & - ⟨ v \nabla_{\vec{β}} ln (1 - F [\vec{β}]) ⟩ = ⟨ v \frac{\nabla_{\vec{β}} F [\vec{β}]}{1 - F [\vec{β}]} ⟩ \\ = & - ⟨ \frac{F [\vec{β}] (1 - F [\vec{β}]) v \otimes \vec{e}}{1 - F [\vec{β}]} ⟩ = ⟨ F [\vec{β}] (1 - F [\vec{β}]) v \otimes \vec{e} ⟩ = \vec{υ} {(\vec{ρ})}^{T} . \end{array}$

Consider the function $τ (\vec{ρ})$ given by $τ (\vec{ρ}) + τ^{*} (\vec{β}) = \vec{β} ⊙ \vec{υ} (\vec{ρ})$ with $\vec{β} = T^{- 1} (\vec{ρ})$ . We can simplify it by writing $τ (\vec{ρ}) = \vec{β} ⊙ \vec{υ} (\vec{ρ}) - τ^{*} (\vec{β}) = ⟨ v ln (1 - F [\vec{β}]) - \vec{β} ⊙ \vec{e} v F [\vec{β}] ⟩ .$ Since $\vec{β} ⊙ \vec{e} = ln (\frac{1}{F} - 1)$ , we can further simplify as $τ (\vec{ρ}) = ⟨ v (1 - F [\vec{β}]) ln (1 - F [\vec{β}]) + v F [\vec{β}] ln (F [\vec{β}]) ⟩ .$ We conclude that τ is the entropy flux of the Fermi–Dirac distribution associated with the conserved densities $\vec{ρ}$ . □

6. Applications to the Eulerian limit Theorem 5.

Assume that the collision operator satisfies properties (2.2) , (2.4) and (2.6). We consider $F_{ε}$ – a sequence of non-negative solutions of $(\partial_{t} + v \cdot \nabla_{x}) F_{ε} = \frac{1}{ε} C (F_{ε}), (t, x, v) \in R_{+} \times R^{3} \times R^{3},$ (6.1) such that $F_{ε} (t, x, v)$ converges to a non-negative function F almost everywhere in $R_{+} \times R^{3} \times R^{3}$ as ε tends to zero. Assume also that the moments $⟨ F_{ε} ⟩, ⟨ v F_{ε} ⟩, ⟨ v \otimes v F_{ε} ⟩, ⟨ | v |^{2} v F_{ε} ⟩$ converge in the sense of distributions to the corresponding moments $⟨ F ⟩, ⟨ v F ⟩, ⟨ v \otimes v F ⟩, ⟨ | v |^{2} v F ⟩,$ that the entropy density and entropy flux converge in the sense of distributions $\begin{array}{rcl} lim_{ε \to 0} ⟨ F_{ε} ln F_{ε} + (1 - F_{ε}) ln (1 - F_{ε}) ⟩ = ⟨ F ln F + (1 - F) ln (1 - F) ⟩, \\ lim_{ε \to 0} ⟨ v F_{ε} ln F_{ε} + v (1 - F_{ε}) ln (1 - F_{ε}) ⟩ = ⟨ v F ln F + v (1 - F) ln (1 - F) ⟩, \end{array}$ and finally that the entropy production rate satisfy $\underset{ε \to 0}{lim inf} ⟨ C (F_{ε}) ln (\frac{1 - F_{ε}}{F_{ε}}) ⟩ ⩾ ⟨ C (F) ln (\frac{1 - F}{F}) ⟩ .$ Then the limit F is a local Fermi–Dirac distribution $F (t, x, v) = F [\nabla_{\vec{ρ}} σ (\vec{ρ} (t, x))],$ where the vector of conserved densities $\vec{ρ} (t, x)$ satisfies the system of conservation laws $\partial_{t} \vec{ρ} + \nabla_{x} \cdot \vec{υ} (\vec{ρ}) = 0$ (6.2) together with the entropy inequality $\partial_{t} σ (\vec{ρ}) + \nabla_{x} \cdot τ (\vec{ρ}) ⩽ 0$ (6.3) in the sense of distributions on $R_{+}^{*} \times R^{3} \times R^{3}$ .

Proof.

Multiplying Eq. (6.1) by $ε ln (\frac{F_{ε}}{1 - F_{ε}})$ and integrating with respect to v gives $\begin{array}{rclr} ε \partial_{t} ⟨ F_{ε} ln F_{ε} + (1 - F_{ε}) ln (1 - F_{ε}) ⟩ + ε \nabla_{x} \cdot ⟨ v F_{ε} ln F_{ε} + v (1 - F_{ε}) ln (1 - F_{ε}) ⟩ \\ ⩽ ⟨ C (F_{ε}) ln (\frac{F_{ε}}{1 - F_{ε}}) ⟩ . & (6.4) \end{array}$ The left-hand side of the above expression tends to zero in the sense of distributions as ε tends to zero. On the other hand, $⟨ C (F_{ε}) ln (\frac{F_{ε}}{1 - F_{ε}}) ⟩ ⩽ 0$ , therefore it is a Radon measure. Hence, $\iint_{R_{+}^{*} \times R^{3}} ⟨ C (F_{ε}) ln (\frac{F_{ε}}{1 - F_{ε}}) ⟩ ϕ (t, x) d x d t \to 0 \forall ϕ \in C_{c} (R_{+}^{*} \times R^{3}) .$ In particular, this holds for $ϕ ⩾ 1_{[a, b] \times B (0, R)}$ , so $\iint_{[a, b] \times B (0, R)} ⟨ C (F_{ε}) ln (\frac{F_{ε}}{1 - F_{ε}}) ⟩ d x d t \to 0,$ by Fatou’s lemma we conclude that $\iint_{[a, b] \times B (0, R)} ⟨ C (F) ln (\frac{F}{1 - F}) ⟩ d x d t \to 0 \forall a, b, \forall R .$ The characterisation of equilibria (3.5) allows us to say that for almost every $(t, x) \in R_{+} \times R^{3}$ the distribution F is a solution of $C (F) = 0$ and has therefore the form (2.5).

In the system of local conservation laws $\begin{matrix} \partial_{t} ⟨ F_{ε} ⟩ + \nabla_{x} \cdot ⟨ v F_{ε} ⟩ = 0, \\ \partial_{t} ⟨ v F_{ε} ⟩ + \nabla_{x} \cdot ⟨ v \otimes v F_{ε} ⟩ = 0, \\ \partial_{t} ⟨ | v |^{2} F_{ε} ⟩ + \nabla_{x} \cdot ⟨ v | v |^{2} F_{ε} ⟩ = 0 \end{matrix}$ (6.5) we pass to the limit in the sense of distributions. Thanks to the convergence assumptions of this theorem, we obtain the compressible Euler system (6.2).

We rewrite (6.4) as $\begin{array}{rclr} \partial_{t} ⟨ F_{ε} ln F_{ε} + (1 - F_{ε}) ln (1 - F_{ε}) ⟩ + \nabla_{x} \cdot ⟨ v F_{ε} ln F_{ε} + v (1 - F_{ε}) ln (1 - F_{ε}) ⟩ \\ ⩽ \frac{1}{ε} ⟨ C (F_{ε}) ln (\frac{F_{ε}}{1 - F_{ε}}) ⟩ . & (6.6) \end{array}$

The right-hand side is non-positive by the characterization (3.5) and the left-hand side converges in the sense of distributions to $\partial_{t} σ (\vec{ρ}) + \nabla_{x} \cdot τ (\vec{ρ})$ , which yields (6.3). □

Observe that the system (6.2) has the convex entropy σ, therefore, it has Godunov’s structure and hence is hyperbolic (see, for example, [7–9] and [10]). The hyperbolicity of the system (6.2) implies several useful properties, notably that it has a unique smooth local solution.

Footnotes

Properties of polylogarithms

We study the polylogarithms for the argument $p > 0$ defined by the formula (A.1) ${Li}_{p} (- e^{w}) = - \frac{1}{Γ (p)} \int_{0}^{\infty} \frac{t^{p - 1}}{e^{t - w} + 1} d t .$ In order to simplify the reasoning, we introduce the notations (A.2) $F_{p} (w) = - {Li}_{p} (- e^{w}), G_{p} (w) = {(F_{p} (w))}^{1 / p} .$ We will use the following properties:

$\frac{d}{d w} F_{p} (w) = F_{p - 1} (w)$ ;

for $p ⩾ 0$ the function $w \to F_{p} (w)$ is positive and monotonically increasing. Moreover, $F_{p} (w) \sim \frac{w^{p}}{Γ (p + 1)}$ as $w \to + \infty$ ;

if $p > 0$ , then $\frac{d^{2}}{d w^{2}} G_{p} (w) > 0 ⟺ \frac{p F_{p} (w)}{F_{p - 1} (w)} > \frac{(p - 1) F_{p - 1} (w)}{F_{p - 2} (w)} .$

Theorem A.1.

If $p ⩾ 2$ , then the function $w \to \frac{d^{2}}{d w^{2}} G_{p} (w)$ is strictly positive.

Proof.

For each fixed w, we examine the function $Ψ (p) = \frac{p F_{p} (w)}{F_{p - 1} (w)} = \frac{\int_{0}^{\infty} t^{p - 1} d μ (t)}{\int_{0}^{\infty} t^{p - 2} d μ (t)},$ with $d μ (t) = \frac{d t}{e^{t - w} + 1} .$ Clearly, we have $t^{s} \in L^{1} (d μ (t))$ for all $s > - 1$ , so that $Ψ (p)$ is defined for $p > 1$ . By continuity, we can put $Ψ (1) = 0$ .

Indeed, $\int_{0}^{\infty} t^{p - 1} d μ (t)$ converges to $\int_{0}^{\infty} d μ (t)$ as $p \to 1$ by Lebesgue theorem; on the other hand, $lim_{p \to 1} \int_{0}^{\infty} t^{p - 2} d μ (t) = + \infty$ by the comparison test, which implies that $lim_{p \to 1, p > 1} Ψ (p) = 0 .$ We want to prove that $Ψ (p) > Ψ (p - 1)$ for $p ⩾ 2$ .

Let us consider the function $Φ : x \to \int_{0}^{\infty} t^{x} d μ (t) .$ We claim that this function is strictly log-convex. Indeed, $Φ > 0$ , so $ln \circ Φ$ is defined.

Let us take $\frac{1}{m} + \frac{1}{n} = 1$ with $m, n \in [1, \infty]$ and $y > x > - 1$ . By Hölder’s inequality, $\int_{0}^{\infty} t^{x / n + y / m} d μ (t) ⩽ {(\int_{0}^{\infty} t^{x} d μ (t))}^{1 / n} {(\int_{0}^{\infty} t^{y} d μ (t))}^{1 / m},$ hence $\begin{array}{rcl} ln (Φ (x / m + y / n)) & = & ln \int_{0}^{\infty} t^{x / m + y / n} d μ (t) \\ < & \frac{1}{m} ln \int_{0}^{\infty} t^{x} d μ (t) + \frac{1}{n} ln \int_{0}^{\infty} t^{y} d μ (t) = \frac{1}{m} ln Φ (x) + \frac{1}{n} ln Φ (y) . \end{array}$ The inequality is strict because the functions $t \to t^{x}$ and $t \to t^{y}$ are linearly independent since $x \neq y$ .

Thus, the function $p \to ln (Φ (p))$ is strictly convex. Clearly, this function is $C^{\infty} (- 1, \infty)$ . In particular, the function $p \to \frac{ln (Φ (p)) - ln (Φ (p - 1))}{p - (p - 1)} = ln (Ψ (p))$ is strictly increasing wherever it is defined. Indeed, let us suppose that its derivative is zero at some point: $0 = ln {(Φ (p))}^{'} - ln {(Φ (p - 1))}^{'} = \int_{p - 1}^{p} ln {(Φ (s))}^{″} d s .$ As $ln {(Φ (s))}^{″} ⩾ 0$ , we conclude that $ln {(Φ (s))}^{″} = 0$ on the interval $[p - 1, p]$ , which contradicts the strict convexity of $ln \circ Φ$ on its domain of definition, hence $ln (Ψ (p))$ is strictly increasing. It follows immediately that Ψ itself is strictly increasing, which assures that $Ψ (p + 1) > Ψ (p)$ on its domain of definition.

Thus, we conclude that $w \to G_{p} (w)$ has a strictly positive second derivative for $p ⩾ 2$ . □

Remark.

We also conjecture that $w \to G_{p} (w)$ is convex for $p ⩾ 1$ , which is supported by numerical evidence and the fact that $G_{1} (w) = ln (1 + e^{w})$ is a strictly convex function.

Technical result

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The Euler limit for kinetic models with Fermi–Dirac statistics

Abstract

Keywords

1. Introduction

Footnotes

Properties of polylogarithms

Technical result

References