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Register a new userPre-asymptotic dynamics of the infinite size Neumann (p=2 spherical) model
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Leticia Cugliandolo
Publication date
2019
fields
Physics
and research's language is
English
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In this contribution we further study the classical disordered p=2 spherical model with Hamiltonian dynamics, or in integrable systems terms, the Neumann model, in the infinite size limit. We summarise the asymptotic results that some of us presented in a recent publication, and we deepen the analysis of the pre-asymptotic dynamics. We also discuss the possible description of the asymptotic steady state with a Generalised Gibbs Ensemble.
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We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body interactions drawn from a Gaussian probability distribution. In the statistical physics framework, the potential energy is of the so-called $p=2$ spherical disordered kind. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable system. We take initial conditions in thermal equilibrium and we subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian. We identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We obtain the {it global} dynamical observables with numerical and analytic methods and we show that, in most cases, they are out of thermal equilibrium. We note, however, that for shallow quenches from the condensed phase the dynamics are close to (though not at) thermal equilibrium. Surprisingly enough, for a particular relation between parameters the global observables comply Gibbs-Boltzmann equilibrium. We next set the analysis of the system with finite number of degrees of freedom in terms of $N$ non-linearly coupled modes. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the $N-1$ integrals of motion and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs-Boltzmann behaviour of global observables. We elaborate on the role played by these constants of motion in the post-quench dynamics and we briefly discuss the possible description of the asymptotic dynamics in terms of a Generalised Gibbs Ensemble.
Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite size $2$-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.
A detailed analysis of the finite-size effects on the bulk critical behaviour of the $d$-dimensional mean spherical model confined to a film geometry with finite thickness $L$ is reported. Along the finite direction different kinds of boundary conditions are applied: periodic $(p)$, antiperiodic $(a)$ and free surfaces with Dirichlet $(D)$, Neumann $(N)$ and a combination of Neumann and Dirichlet $(ND)$ on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary $d$. It is found, for $2<d<4$, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for $(p)$ and $(a)$. For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At $d=3$, the critical amplitude of the singular part of the free energy (related to the so called Casimir amplitude) is estimated. We obtain $Delta^{(p)}=-2zeta(3)/(5pi)=-0.153051...$, $Delta^{(a)}=0.274543...$ and $Delta^{(ND)}=0.01922...$, implying a fluctuation--induced attraction between the surfaces for $(p)$ and repulsion in the other two cases. For $(D)$ and $(N)$ we find a logarithmic dependence on $L$.
We revisit the gradient descent dynamics of the spherical Sherrington-Kirkpatrick ($p=2$) model with finite number of degrees of freedom. For fully random initial conditions we confirm that the relaxation takes place in three time regimes: a first algebraic one controlled by the decay of the eigenvalue distribution of the random exchange interaction matrix at its edge in the infinite size limit; a faster algebraic one determined by the distribution of the gap between the two extreme eigenvalues; and a final exponential one determined by the minimal gap sampled in the disorder average. We also analyse the finite size effects on the relaxation from initial states which are almost projected on the saddles of the potential energy landscape, and we show that for deviations scaling as $N^{- u}$ from perfect alignment the system escapes the initial configuration in a time-scale scaling as $ln N$ after which the dynamics no longer self-averages with respect to the initial conditions. We prove these statements with a combination of analytic and numerical methods.
We study Kleinberg navigation (the search of a target in a d-dimensional lattice, where each site is connected to one other random site at distance r, with probability proportional to r^{-a}) by means of an exact master equation for the process. We show that the asymptotic scaling behavior for the delivery time T to a target at distance L scales as (ln L)^2 when a=d, and otherwise as L^x, with x=(d-a)/(d+1-a) for a<d, x=a-d for d<a<d+1, and x=1 for a>d+1. These values of x exceed the rigorous lower-bounds established by Kleinberg. We also address the situation where there is a finite probability for the message to get lost along its way and find short delivery times (conditioned upon arrival) for a wide range of as.
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