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First-principle validation of Fourier’s law in d=1,2,3 classical systems

We numerically study the thermal transport in the classical inertial nearest-neighbor XY ferromagnet in d=1,2,3, the total number of sites being given by N=Ld, where L is the linear size of the system.

For the thermal conductance σ, we obtain σ(T,L)Lδ(d)=A(d)eB(d)[Lγ(d)T]η(d)q(d) (with ezq[1+(1q)z]1/(1q);ez1=ez;A(d)>0;B(d)>0;q(d)>1;η(d)>2;δ0;γ(d)>0), for all values of Lγ(d)T for d=1,2,3. In the L limit, we have σ1/Lρσ(d) with ρσ(d)=δ(d)+γ(d)η(d)/[q(d)1]. The material conductivity is given by κ=σLd1/Lρκ(d) (L) with ρκ(d)=ρσ(d)d.

Our numerical results are consistent with ‘conspiratory’ d-dependences of (q,η,δ,γ), which comply with normal thermal conductivity (Fourier law) for all dimensions.

C. Tsallis, H. S. Lima, U. Tirnakli, D. Eroglu, First-principle validation of Fourier’s law in d=1,2,3 classical systems, Physica D: Nonlinear Phenomena.

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