Fluctuations in energy in the one-dimensional crystal with a substrate

[math] \def\be#1{\begin{equation}\label{#1}} \def\ee{\end{equation}} \def\({\left(} \def\){\right)} \let\eps=\varepsilon \let\w=\omega \let\al=\alpha \renewcommand {\=}{\mathrel{\stackrel{\rm def}=}} [/math] Examines the chain consisting of identical masses [math]m[/math], connected by springs stiffness [math]C_0[/math]. The chain is on an elastic foundation stiffness [math]C_1[/math]. Then the equation of the dynamics of the chain of particles is of the form:

[math] \be{1Delta} \ddot{u}_n =\(\w^2_0 \Delta^2_n-\w^2_1\) u_n ,\qquad \w_0\=\sqrt{C_0/m} ,\qquad \w_1\=\sqrt{C_1/m} ,\ee [/math]

where [math]u_n[/math] — moving the [math]n[/math]-th particle; [math]\Delta^2_n[/math] — difference operator of second order:

[math] \be{delta2} \Delta^2_n u_n \= u_{n-1}-2u_{n}+u_{n+1}, \ee [/math]

[math]n[/math] — an index that takes arbitrary integer.

Developers: Tsvetkov Denis, Krivtsov Anton