Upper bound on the decay of correlations in the plane rotator model with long-range random interaction

We give an upper bound on the decay of correlation function for the plane rotator model with Hamiltonian $$ - \frac{1}{2}\mathop \sum \limits_{xy} \frac{{J_{xy} \cos (\theta _x - \theta _y )}}{{\| {x - y} \|^{({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2} + \varepsilon )^d } }}$$ in dimensiond=1 andd=2 when (J _xy are independent random variables with mean zero.     

On the freund-witten adelic formula for Veneziano amplitudes

On the basis of the analysis of the adele group (Tate's formula), a regularization for the divergent infinite product ofp-adic Г-functions $$\Gamma _p (\alpha ) = \frac{{1 - p^{\alpha - 1} }}{{[ - p^{ - \alpha } }}$$ is proposed, and the adelic formula is proved $$reg\coprod\limits_{p = 2}^\infty {\Gamma _p (\alpha )} = \frac{{\zeta (\alpha )}}{{\zeta (1 - \alpha )}}$$ whereζ(α) is the Riemannζ-function.     

Path integral treatment of a noncentral electric potential

We present a rigorous path integral treatment of a dynamical system in the axially symmetric potential $V(r,\theta ) = V(r) + \tfrac{1} {{r^2 }}V(\theta ) $ . It is shown that the Green’s function can be calculated in spherical coordinate system for $V(\theta ) = \frac{{\hbar ^2 }} {{2\mu }}\frac{{\gamma + \beta \sin ^2 \theta + \alpha \sin ^4 \theta }} {{\sin ^2 \theta \cos ^2 \theta }} $ . As an illustration, we have chosen the example of a spherical harmonic oscillator and also the Coulomb potential for the radial dependence of this noncentral potential. The ring-shaped oscillator and the Hartmann ring-shaped potential are considered as particular cases. When α = β = γ = 0, the discrete energy spectrum, the normalized wave function of the spherical oscillator and the Coulomb potential of a hydrogen-like ion, for a state of orbital quantum number l ≥ 0, are recovered.     

The magnetic susceptibility of dilute CuFe alloys in the temperature range 1.7–300 K

The paramagnetic susceptibility ofCuFe alloys (100–300 ppm Fe) has been measured in the temperature range 1.7 to 300 K. The susceptibility is very well represented by the expression $$\chi = \chi _0 + \frac{{C_1 }}{{T - \theta _1 }} + \frac{{C_2 }}{{T - \theta _2 }}$$ the first and second Curie Weiss terms being associated with single and paired iron atoms respectively. A Cu_0-9 Au_0-1 Fe alloy was also studied.     

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

One-range Addition Theorems for Noninteger n Slater Functions using Complete Orthonormal Sets of Exponential Type Orbitals in Standard Convention

By the use of complete orthonormal sets of ${\psi ^{(\alpha^{\ast})}}$ -exponential type orbitals ( ${\psi ^{(\alpha^{\ast})}}$ -ETOs) with integer (for α ^* = α) and noninteger self-frictional quantum number α ^*(for α ^* ≠ α) in standard convention introduced by the author, the one-range addition theorems for ${\chi }$ -noninteger n Slater type orbitals ${(\chi}$ -NISTOs) are established. These orbitals are defined as follows $$\begin{array}{ll}\psi _{nlm}^{(\alpha^*)} (\zeta ,\vec {r}) = \frac{(2\zeta )^{3/2}}{\Gamma (p_l ^* + 1)} \left[{\frac{\Gamma (q_l ^* + )}{(2n)^{\alpha ^*}(n - l - 1)!}} \right]^{1/2}e^{-\frac{x}{2}}x^{l}_1 F_1 ({-[ {n - l - 1}]; p_l ^* + 1; x})S_{lm} (\theta ,\varphi )\\ \chi _{n^*lm} (\zeta ,\vec {r}) = (2\zeta )^{3/2}\left[ {\Gamma(2n^* + 1)}\right]^{{-1}/2}x^{n^*-1}e^{-\frac{x}{2}}S_{lm}(\theta ,\varphi ),\end{array}$$ where ${x=2\zeta r, 0<\zeta <\infty , p_l ^{\ast}=2l+2-\alpha ^{\ast}, q_l ^{\ast}=n+l+1-\alpha ^{\ast}, -\infty <\alpha ^{\ast} <3 , -\infty <\alpha \leq 2,_1 F_1 }$ is the confluent hypergeometric function and ${S_{lm} (\theta ,\varphi )}$ are the complex or real spherical harmonics. The origin of the ${\psi ^{(\alpha ^{\ast})} }$ -ETOs, therefore, of the one-range addition theorems obtained in this work for ${\chi}$ -NISTOs is the self-frictional potential of the field produced by the particle itself. The obtained formulas can be useful especially in the electronic structure calculations of atoms, molecules and solids when Hartree–Fock–Roothan approximation is employed.     

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: $$ i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)\quad \varepsilon > 0\,\quad V\in L^1(\mathbb{R},(1+|x|){\rm d}x) \cap L^\infty(\mathbb{R}).$$ This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit \({\varepsilon\to 0}\) the weak (integral) dynamics converges in \({H^1(\mathbb{R})}\) to the weak dynamics of the NLS with point-concentrated nonlinearity: $$ i\frac{{\rm d}}{{\rm d}t} \psi(t) =H_{\alpha} \psi(t) .$$ where H _α is the Laplacian with the nonlinear boundary condition at the origin \({\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)}\) and \({\alpha=\int_{\mathbb{R}}V{\rm d}x}\) . The convergence occurs for every \({\mu\in \mathbb{R}^+}\) if V ≥  0 and for every  \({\mu\in (0,1)}\) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points.     

Phase transition of the symmetric Ashkin-Teller model near the decoupling limit

The Ashkin-Teller model can be viewed as two ferromagnetic Ising models interacting via a four-spin coupling ε. We investigate the phase transition of the symmetric AT-model near the decoupling limit ε=0 and compare it with the 8-Vertex model phase transition. The expansion of the free energy is derived to second order in ε. From this the specific heat exponent is determined to the same order with the result: $$\alpha (\varepsilon ) = \frac{8}{\pi }.\varepsilon + \frac{{16}}{\pi }\left( {1 - \frac{4}{\pi }} \right)\varepsilon ^2 + O(\varepsilon ^3 ).$$      

Three-loop on-shell charge renormalization without integration:\Lambda _{QED}^{\overline {MS} } to four loops

Using algebraic methods, we find the three-loop relation between the bare and physical couplings of one-flavourD-dimensional QED, in terms of Γ functions and a singleF ₃₂ series, whose expansion nearD=4 is obtained, by wreath-product transformations, to the order required for five-loop calculations. Taking the limitD→4, we find that the \(\overline {MS} \) coupling \(\bar \alpha (\mu )\) satisfies the boundary condition $$\begin{gathered} \frac{{\bar \alpha (m)}}{\pi } = \frac{\alpha }{\pi } + \frac{{15}}{{16}}\frac{{\alpha ^3 }}{{\pi ^3 }} + \left\{ {\frac{{11}}{{96}}\zeta (3) - \frac{1}{3}\pi ^2 \log 2} \right. \hfill \\ \left. { + \frac{{23}}{{72}}\pi ^2 - \frac{{4867}}{{5184}}} \right\}\frac{{\alpha ^4 }}{{\pi ^4 }} + \mathcal{O}(\alpha ^5 ), \hfill \\ \end{gathered} $$ wherem is the physical lepton mass and α is the physical fine structure constant. Combining this new result for the finite part of three-loop on-shell charge renormalization with the recently revised four-loop term in the \(\overline {MS} \) β-function, we obtain $$\begin{gathered} \Lambda _{QED}^{\overline {MS} } \approx \frac{{me^{3\pi /2\alpha } }}{{(3\pi /\alpha )^{9/8} }}\left( {1 - \frac{{175}}{{64}}\frac{\alpha }{\pi } + \left\{ { - \frac{{63}}{{64}}\zeta (3)} \right.} \right. \hfill \\ \left. { + \frac{1}{2}\pi ^2 \log 2 - \frac{{23}}{{48}}\pi ^2 + \frac{{492473}}{{73728}}} \right\}\left. {\frac{{\alpha ^2 }}{{\pi ^2 }}} \right), \hfill \\ \end{gathered} $$ at the four-loop level of one-flavour QED.     

On the absence of breakdown of symmetry for the plane rotator model with long-range random interaction

We study the plane rotator model with hamiltonian $$ - \frac{1}{2}\sum\limits_{x \ne y} {J_{xy} \frac{{\cos (\theta _x - \theta _y )}}{{\left| {\left. {x - y} \right|} \right.^{3 + \in } }}}$$ whereJ _xy for different pair (x, y) are independent symmetric random variables. It is proved that for almost allJ, all the Gibbs statesP(J) are rotation invariant.     

On the similarity solution of evolution equation u t =H(x,t, u,u x,u xx , ...)

Let $$\begin{gathered} u^* = u + \in \eta (x,{\text{ }}t,{\text{ }}u), \hfill \\ \hfill \\ \hfill \\ x^* = x + \in \xi (x, t, u{\text{),}} \hfill \\ \hfill \\ \hfill \\ {\text{t}}^{\text{*}} = {\text{ }}t + \in \tau {\text{(}}x,{\text{ }}t,{\text{ }}u), \hfill \\ \end{gathered}$$ be an infinitesimal invariant transformation of the evolution equation u _t =H(x,t,u,?u/?x,...,? ⁿ :u/?x ⁿ . In this paper we give an explicit expression for \(\eta ^{X^i }\) in the ‘determining equation’ $$\eta ^T = \sum\limits_{i = 1}^n {{\text{ }}\eta ^{X^i } {\text{ }}\frac{{\partial H}}{{\partial u_i }} + \eta \frac{{\partial H}}{{\partial u_{} }} + \xi \frac{{\partial H}}{{\partial x}} + \tau } \frac{{\partial H}}{{\partial t}},$$ where u _i =? ⁱ u/?x ⁱ . By using this expression we derive a set of equations with η, ξ, τ as unknown functions and discuss in detail the cases of heat and KdV equations.     

On relativistic deformable solids and the detection of gravitational waves

In this paper, the purpose of which is to complement a preceding work [1], it is shown, in agreement with the theory of relativistic deformable solids developed by A.C. Bringen and his coworkers, that the simplest conceivable dissipative constitutive equation — that of a socalled KelvinVoigt viscoelastic solid — yields a gravitational wave equation of propagation different from that of Weber: specifically, the following third order partial differential equation, $$\frac{{\partial ^2 \theta }}{{\partial t^2 }} - \left( {A + A'\frac{{\partial ^2 \theta }}{{\partial t}}} \right)\frac{{\partial ^2 \theta }}{{\partial x^2 }} = c^2 R_{1441'} $$ which can be solved by use of Fourier transform techniques, and where A and A′ are positive material coefficients.     

Multiphonon thermal-field ionization of deep centers in semiconductors

The problem of thermal-field ionization of deep impurity centers in semiconductors is studied. It is shown that \(W_{ion} = W_0 e^{\alpha F^2 }\) , where F is the electric field strength. Also, the lifetime for multiphonon nonradiative capture is calculated as a function of F. It is shown that the relative change in lifetime is $$\frac{{\Delta \tau }}{{\tau ^0 }} = \frac{{\tau ---\tau _0 }}{{\tau _0 }} \approx - \alpha F^2 .$$      

BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one

We show that for most non-scalar systems of conservation laws in dimension greater than one, one does not have BV estimates of the form $$\begin{gathered} \parallel \overline V u(\overline t )\parallel _{T.V.} \leqq F(\parallel \overline V u(0)\parallel _{T.V.} ), \hfill \\ F \in C(\mathbb{R}),F(0) = 0,F Lipshitzean at 0, \hfill \\ \end{gathered} $$ even for smooth solutions close to constants. Analogous estimates forL ^p norms $$\parallel u(\overline t ) - \overline u \parallel _{L^p } \leqq F(\parallel u(0) - \overline u \parallel _{L^p } ),p \ne 2$$ withF as above are also false. In one dimension such estimates are the backbone of the existing theory.     

Asymptotic Analysis of the Spatially Homogeneous Boltzmann Equation: Grazing Collisions Limit

In the present work, we consider the asymptotic problem of the spatially homogeneous Boltzmann equation when almost all collisions are grazing, that is, the deviation angle $\theta $ of the collision is limited near zero (i.e., $\theta \le \epsilon $ ). We show that by taking the proper scaling to the cross-section which was used in [37], that is, assuming $$\begin{aligned} B^\epsilon ( v-v_{*},\sigma )=2(1-s)|v-v_*|^{\gamma }\epsilon ^{-3}\sin ^{-1}\theta \left( \frac{\theta }{\epsilon }\right) ^{-1-2s}\mathrm {1}_{\theta \le \epsilon }, \end{aligned}$$ where $\theta = \langle \theta ={\frac{\upsilon -\upsilon _*}{|\upsilon -\upsilon _*|}}.\sigma \rangle , $ the solution $f^\epsilon $ of the Boltzmann equation with initial data $f_0$ can be globally or locally expanded in some weighted Sobolev space as $$\begin{aligned} f^\epsilon = f+ O(\epsilon ), \end{aligned}$$ where the function $f$ is the solution of Landau equation, which is associated with the grazing collisions limit of Boltzmann equation, with the same initial data $f_0$ . This gives the rigorous justification of the Landau approximation in the spatially homogeneous case. In particular, if taking $\gamma =-3$ and $s=1-\epsilon $ in the cross-section $B^\epsilon $ , we show that the above asymptotic formula still holds and in this case $f$ is the solution of Landau equation with the Coulomb potential. Going further, we revisit the well-posedness problem of the Boltzmann equation in the limiting process. We show there exists a common lifespan such that the uniform estimates of high regularities hold for each solution $f^\epsilon $ . Thanks to the weak convergence results on the grazing collisions limit in [37], in other words, we establish a unified framework to establish the well-posedness results for both Boltzmann and Landau equations.     

Boundedness of total cross-sections in potential scattering. II

If, in addition to the condition $$\frac{1}{{(4\pi )^2 }}\int {d^3 xd^3 x'} \frac{{|V(x)||V(x')|}}{{|x - x'|^2 }}< 1$$ in units where 2M/?² = 1, which guarantees that the total cross-section averaged over incident directions is finite, we have also $$\frac{1}{{(4\pi )}}\int {d^3 xd^3 x'} \frac{{|V(x)||V(x')|}}{{|x - x'|}}$$ finite, the total cross-section is finite for all energies and all directions of the incident beam.     

Inelastic Character of Solitons of Slowly Varying gKdV Equations

In this paper we study soliton-like solutions of the variable coefficients, the subcritical gKdV equation $$u_t + (u_{xx} -\lambda u + a(\varepsilon x) u^m )_x =0,\quad {\rm in} \quad \mathbb{R}_t\times\mathbb{R}_x, \quad m=2,3\,\, { \rm and }\,\, 4,$$ with ${\lambda\geq 0, a(\cdot ) \in (1,2)}$ a strictly increasing, positive and asymptotically flat potential, and ${\varepsilon}$ small enough. In previous works (Mu?oz in Anal PDE 4:573?C638, 2011; On the soliton dynamics under slowly varying medium for generalized KdV equations: refraction vs. reflection, SIAM J. Math. Anal. 44(1):1?C60, 2012) the existence of a pure, global in time, soliton u(t) of the above equation was proved, satisfying $$\lim_{t\to -\infty}\|u(t) - Q_1(\cdot -(1-\lambda)t) \|_{H^1(\mathbb{R})} =0,\quad 0\leq \lambda<1,$$ provided ${\varepsilon}$ is small enough. Here R(t, x) := Q _c (x ? (c ? ??)t) is the soliton of R _t +? (R _xx ??? R + R ^m ) _x =?0. In addition, there exists ${\tilde \lambda \in (0,1)}$ such that, for all 0?<??? <?1 with ${\lambda\neq \tilde \lambda}$ , the solution u(t) satisfies $$\sup_{t\gg \frac{1}{\varepsilon}}\|u(t) - \kappa(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}\lesssim \varepsilon^{1/2}.$$ Here ${{\rho'(t) \sim (c_\infty(\lambda) -\lambda)}}$ , with ${{\kappa(\lambda)=2^{-1/(m-1)}}}$ and ${{c_\infty(\lambda)>\lambda}}$ in the case ${0<\lambda<\tilde\lambda}$ (refraction), and ${\kappa(\lambda) =1}$ and c _??(??)?<??? in the case ${\tilde \lambda<\lambda<1}$ (reflection). In this paper we improve our preceding results by proving that the soliton is far from being pure as t ?? +???. Indeed, we give a lower bound on the defect induced by the potential a(·), for all ${{0<\lambda<1, \lambda\neq \tilde \lambda}}$ . More precisely, one has $$\liminf_{t\to +\infty}\| u(t) - \kappa_m(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}>rsim \varepsilon^{1 +\delta},$$ for any ${{\delta>0}}$ fixed. This bound clarifies the existence of a dispersive tail and the difference with the standard solitons of the constant coefficients, gKdV equation.     

Level spacing distributions and the Bessel kernel

Scaling models of randomN×N hermitian matrices and passing to the limitN→∞ leads to integral operators whose Fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order. For the Gaussian Unitary Ensemble, and for many others'as well, the kernel one obtains by scaling in the “bulk” of the spectrum is the “sine kernel” $\frac{{\sin \pi (x - y)}}{{\pi (x - y)}}$ . Rescaling the GUE at the “edge” of the spectrum leads to the kernel $\frac{{Ai(x)Ai'(y) - Ai'(x)Ai(y)}}{{x - y}}$ , where Ai is the Airy function. In previous work we found several analogies between properties of this “Airy kernel” and known properties of the sine kernel: a system of partial differential equations associated with the logarithmic differential of the Fredholm determinant when the underlying domain is a union of intervals; a representation of the Fredholm determinant in terms of a Painlevé transcendent in the case of a single interval; and, also in this case, asymptotic expansions for these determinants and related quantities, achieved with the help of a differential operator which commutes with the integral operator. In this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the Laguerre or Jacobi ensembles at the edge of the spectrum, namely $$\frac{{J_\alpha (\sqrt x )\sqrt y J'_\alpha (\sqrt y ) - \sqrt x J'_\alpha (\sqrt x )J_\alpha (\sqrt y )}}{{2(x - y)}},$$ , whereJ _α(z) is the Bessel function of order α. In the cases α=?1/2 these become, after a variable change, the kernels which arise when taking scaling limits in the bulk of the spectrum for the Gaussian orthogonal and symplectic ensembles. In particular, an asymptotic expansion we derive will generalize ones found by Dyson for the Fredholm determinants of these kernels.     

An analytical expression for the specific capacity of lithium ion cells

The concentration of lithium ions in the cathode of lithium ion cells has been obtained by solving the materials balance equation $$\frac{{\partial c}}{{\partial t}} = \varepsilon ^{1/2} D\frac{{\partial ^2 c}}{{\partial x^2 }} + \frac{{aj_n (1--t_ + )}}{\varepsilon }$$ by Laplace transform. On the assumption that the cell is fully discharged when there are zero lithium ions at the current collector of the cathode, the discharge timet _d is obtained as $$\tau = \frac{{r^2 }}{{\pi ^2 \varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{{r^2 }}\left( {\frac{{\varepsilon ^{1/2} }}{J} + \frac{{r^2 }}{6}} \right)} \right]$$ which, when substituted into the equationC=It _d /M, whereI is the discharge current andM is the mass of the separator and positive electrode, an analytical expression for the specific capacity of the lithium cell is given as $$C = \frac{{IL_c ^2 }}{{\pi {\rm M}D\varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{2}\left( {\frac{{FDc_0 \varepsilon ^{3/2} }}{{I(1 - t_ + )L_c }} + \frac{1}{6}} \right)} \right]$$      

Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law

In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the following steady problem: $$v \cdot \nabla _{x}F =\frac{1}{{\rm K}_{\rm n}}Q(F,F),\qquad (x,v)\in \Omega \times \mathbf{R}^{3}, \quad \quad (0.1) $$ v · ? x F = 1 K n Q ( F , F ) , ( x , v ) ∈ Ω × R 3 , ( 0.1 ) $$F(x,v)|_{n(x)\cdot v<0} = \mu _{\theta}\int_{n(x) \cdot v^{\prime}>0}F(x,v^{\prime})(n(x)\cdot v^{\prime})dv^{\prime},\quad x \in\partial \Omega,\quad \quad (0.2) $$ F ( x , v ) | n ( x ) · v < 0 = μ θ ∫ n ( x ) · v ′ > 0 F ( x , v ′ ) ( n ( x ) · v ′ ) d v ′ , x ∈ ? Ω , ( 0.2 ) where Ω is a bounded domain in ${\mathbf{R}^{d}, 1 \leq d \leq 3}$ R d , 1 ≤ d ≤ 3 , K_n is the Knudsen number and ${\mu _{\theta}=\frac{1}{2\pi \theta ^{2}(x)} {\rm exp} [-\frac{|v|^{2}}{2\theta (x)}]}$ μ θ = 1 2 π θ 2 ( x ) exp [ - | v | 2 2 θ ( x ) ] is a Maxwellian with non-constant(non-isothermal) wall temperature θ(x). Based on new constructive coercivity estimates for both steady and dynamic cases, for ${|\theta -\theta_{0}|\leq \delta \ll 1}$ | θ - θ 0 | ≤ δ ? 1 and any fixed value of K_n, we construct a unique non-negative solution F _s to (0.1) and (0.2), continuous away from the grazing set and exponentially asymptotically stable. This solution is a genuine non-equilibrium stationary solution differing from a local equilibrium Maxwellian. As an application of our results we establish the expansion ${F_s=\mu_{\theta_0}+\delta F_{1}+O(\delta ^{2})}$ F s = μ θ 0 + δ F 1 + O ( δ 2 ) and we prove that, if the Fourier law holds, the temperature contribution associated to F ₁ must be linear, in the slab geometry.