Heat transfer in a 1D harmonic crystal

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Theory: A.M. Krivtsov, published at arXiv:1509.02506 (cond-mat.stat-mech)

Programming: D.V. Tsvetkov

Microscopic model

We consider a one-dimensional crystal, described by the following equations of motion:

[math] \ddot{u}_i = \omega_0^2(u_{i-1}-2u_i+u_{i+1}) ,\qquad \omega_0 = \sqrt{C/m}, [/math]

where [math]u_i[/math] is the displacement of the [math]i[/math]th particle, [math]m[/math] is the particle mass, [math]C[/math] is the stiffness of the interparticle bond. The crystal is infinite: the index [math]i[/math] is an arbitrary integer. The initial conditions are

[math] u_i|_{t=0} = 0 ,\qquad \dot u_i|_{t=0} = \sigma(x)\varrho_i , [/math]

where [math]\varrho_i[/math] are independent random values with zero expectation and unit variance; [math]\sigma[/math] is variance of the initial velocities of the particles, which is a slowly varying function of the spatial coordinate [math]x=ia[/math], where [math]a[/math] is the lattice constant. These initial conditions correspond to an instantaneous temperature perturbation, which can be induced in crystals, for example, by an ultrashort laser pulse.

Kinetic temperature: link between micro and macro

The kinetic temperature [math]T[/math] is defined as

[math] T(x) = \frac m{k_{B}}\langle\dot u_i^2\rangle, [/math]

where [math]k_{B}[/math] is the Boltzmann constant, [math]i=x/a[/math], angle brackets stand for mathematical expectation.

Macroscopic equations

— Heat (Fourier): [math]\dot T = \beta T''[/math] [1]

— Heat wave (MCV): [math]\ddot T +\frac1\tau\dot T = \frac\beta\tau T''[/math]

— Wave (d’Alembert): [math]\ddot T = c^2 T''[/math] [2]

— Reversible (Krivtsov): [math]\ddot T +\frac1t\dot T = c^2 T''[/math] [3]

Notations: [math]t[/math] is time (variable), [math]\tau[/math] is the relaxation time (constant), [math]\beta[/math] is the thermal diffusivity, [math]\kappa[/math] is the thermal conductivity, [math]c[/math] is the sound speed, [math]\rho[/math] is the density, MCV stands for Maxwell-Cattaneo-Vernotte.

See also

• Heat transfer in a 1D harmonic crystal: periodic temperature
• Heat transfer in a 1D harmonic crystal: regular temperature