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A note on Harris' ergodic theorem, controllability and perturbations of harmonic networks. (arXiv:1801.05375v2 [math-ph] UPDATED)
来源于:arXiv
We show that elements of control theory, together with an application of
Harris' ergodic theorem, provide an alternate method for showing exponential
convergence to a unique stationary measure for certain classes of networks of
quasi-harmonic classical oscillators coupled to heat baths. With the system of
oscillators expressed in the form $\mathrm{d} X_t = A X_t \,\mathrm{d} t +
F(X_t) \,\mathrm{d} t + B \,\mathrm{d} W_t$ in $\mathbf{R}^d$, where $A$
encodes the harmonic part of the force and $-F$ corresponds to the gradient of
the anharmonic part of the potential, the hypotheses under which we obtain
exponential mixing are the following: $A$ is dissipative, the pair $(A,B)$
satisfies the Kalman condition, $F$ grows sufficiently slowly at infinity
(depending on the dimension $d$), and the vector fields in the equation of
motion satisfy the weak H\"ormander condition in at least one point of the
phase space. 查看全文>>