Abstract
We consider an unbounded lattice and at each point of this lattice an anharmonic oscillator, that interacts with its first neighborhoods via a pair potential V and is subjected to a restoring force of potential U. We assume that U and V are even nonnegative polynomials of degree \(2\sigma _1\) and \(2\sigma _2\). We study the time evolution of this system, with a control of the growth in time of the local energy, and we give a nontrivial bound on the velocity of propagation of a perturbation. This is an extension to the case \(\sigma _1 < 2\sigma _2-1\) of some already known results obtained for \(\sigma _1 \ge 2\sigma _2-1\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
To see this, let \(x^n(t)\) be the time evolution of the oscillators inside the cube \(\Lambda _{0,n}\), assuming empty boundary conditions. Then, for any \(\Lambda _{\mu ,k}\subset \Lambda _{0,n}\) and recalling (2.4),
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} W_{\mu ,k}(x^n(t)) \lesssim \underbrace{W_{\mu ,k}(x^n(t))^{\frac{1}{2}}}_{\text {velocity}} \times \underbrace{W_{\mu ,k+1}(x^n(t))^{\frac{2\sigma _2-1}{2\sigma }}}_{\mathrm {force}}, \end{aligned}$$which can be shown to imply - we omit the details - that
$$\begin{aligned} W_k^n(t) \le W_k^n(0) + C \int _0^t\! W_k^n(s)^{\frac{1}{2} + \frac{2\sigma _2-1}{2\sigma }} \mathrm {d}s, \end{aligned}$$where \(W_k^n(t) := \displaystyle \max _{\mu :\Lambda _{\mu ,k}\subset \Lambda _{0,n}} W_{\mu ,k}(x^n(t))\). If \(\frac{1}{2} + \frac{2\sigma _2-1}{2\sigma } \le 1\), i.e., \(\sigma _1\ge 2\sigma _2-1\), the above inequality can be solved globally in time, getting
$$\begin{aligned} W_k^n(t) \le C(t) W_k^n(0) \le C(t) Q(x) \big [\log (\mathrm {e}+ n)+(2k+1)^d\big ], \end{aligned}$$where we used (2.5), hence an a priori bound on \(W_k^n(t)\), weakly depending on the size n of the finite approximation.
More precisely, the Gibbs distribution of the positions is a Gaussian measure with covariance operator \(S^{-1}\), where S is the force matrix of the harmonic interactions. The finiteness of the variance of each \(q_i\) thus depends on the convergence of the integral \(\int _\Gamma \! \Vert \hat{S}(k)^{-1}\Vert \mathrm {d}k\), where \(\hat{S}(k)\) is the Fourier transformation of S and \(\Gamma \) is the first Brillouin zone. As \(\hat{S}(k)\) is an even function, it vanishes at least as fast as \(|k|^2\) as \(k\rightarrow 0\), so that the variance of \(q_i\) diverges in one and two dimensions. Clearly, one could look only at the random variables which are the differences between nearest neighbor oscillators, but it remains the problem of relating this measure to the evolution of the \(q_i\).
References
Bahan, C., Park, Y.M., Yoo, H.J.: Non equilibrium dynamics of infinite particle systems with infinite range interaction. J. Math. Phys. 40, 4337–4358 (1999)
Benfatto, G., Marchioro, C., Presutti, E., Pulvirenti, M.: Superstability estimates for anharmonic systems. J. Stat. Phys. 22, 349–362 (1980)
Buttà, P., Caglioti, E., Marchioro, C.: On the motion of a charged particle interacting with an infinitely extended system. Commun. Math. Phys. 233, 545–569 (2003)
Buttà, P., Caglioti, E., Marchioro, C.: On the violation of Ohm’s law for bounded interactions: a one dimensional system. Commun. Math. Phys. 249, 353–382 (2004)
Buttà, P., Manzo, F., Marchioro, C.: A simple Hamiltonian model of runaway particle with singular interaction. Math. Model. Methods. Appl. Sci. 15, 753–766 (2005)
Buttà, P., Caprino, S., Cavallaro, G., Marchioro, C.: On the dynamics of infinitely many particles with magnetic confinement. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 9, 371–395 (2006)
Buttà, P., Caglioti, E., Di Ruzza, S., Marchioro, C.: On the propagation of a perturbation in an anharmonic system. J. Stat. Phys. 127, 313–325 (2007)
Buttà, P., Cavallaro, G., Marchioro, C.: Time evolution of two dimensional systems with infinitely many particles mutually interacting via very singular forces. J. Stat. Phys. 147, 412–423 (2012)
Buttà, P., Cavallaro, G., Marchioro, C.: Dynamics of infinitely extended hard core systems. Rep. Math. Phys. 72, 369–377 (2013)
Buttà, P., Cavallaro, G., Marchioro, C.: Mathematical models of viscous friction. In: Lecture Notes in Mathematics, vol. 2135. Springer, Cham (2015)
Caglioti, E., Marchioro, C.: On the long time behavior of a particle in an infinitely extended system in one dimension. J. Stat. Phys. 106, 663–680 (2002)
Caglioti, E., Marchioro, C., Pulvirenti, M.: Non-equilibrium dynamics of three-dimensional infinite particle systems. Commun. Math. Phys. 215, 25–43 (2000)
Caprino, S., Cavallaro, G., Marchioro, C.: Time evolution of an infinitely extended Vlasov fluid with singular mutual interactions. J. Stat. Phys. 162, 426–456 (2016)
Cavallaro, G., Marchioro, C., Spitoni, C.: Dynamics of infinitely many particles mutually interacting in three dimensions via a bounded superstable long-range potential. J. Stat. Phys. 120, 367–416 (2005)
Dobrushin, R.L., Fritz, J.: Non equilibrium dynamics of one-dimensional infinite particle system with hard-core interaction. Commun. Math. Phys. 55, 275–292 (1977)
Fritz, J., Dobrushin, R.L.: Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction. Commun. Math. Phys. 57, 67–81 (1977)
Lanford, O.E., Lebowitz, J.L.: Time evolution and ergodic properties of harmonic systems. In: Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974). Lecture Notes in Physics, vol. 38, pp. 144–177. Springer, Berlin (1975)
Lanford, O.E., Lebowitz, J.L., Lieb, E.: Time evolution of infinite anharmonic systems. J. Stat. Phys. 16, 453–461 (1977)
Lieb, E., Robinson, D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)
Marchioro, C., Pellegrinotti, A., Pulvirenti, M., Triolo, L.: Velocity of a perturbation in infinite lattice systems. J. Stat. Phys. 19, 499–510 (1978)
Marchioro, C., Pellegrinotti, A., Pulvirenti, M.: On the dynamics of infinite anharmonic systems. J. Math. Phys. 22, 1740–1745 (1981)
Acknowledgments
Work performed under the auspices of the Italian Ministry of the University (MIUR).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Buttà, P., Marchioro, C. Dynamics of Infinite Classical Anharmonic Crystals. J Stat Phys 164, 680–692 (2016). https://doi.org/10.1007/s10955-016-1540-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-016-1540-x