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Physical review. E

Finite-size scaling of quasi-stationary-state temperature

Antonio Rodríguez, Fernando Dantas Nobre, Constantino Tsallis

Published Apr 7, 2022• 3 citations
Publisher

Abstract

We numerically study, from first principles, the temperature ${T}_{\mathrm{QSS}}$ and duration ${t}_{\mathrm{QSS}}$ of the longstanding initial quasi-stationary state of the isolated $d$-dimensional classical inertial $\ensuremath{\alpha}\text{\ensuremath{-}}XY$ ferromagnet with two-body interactions decaying as $1/{r}_{ij}^{\ensuremath{\alpha}}$ $(\ensuremath{\alpha}\ensuremath{\ge}0)$. It is shown that this temperature ${T}_{\mathrm{QSS}}$ (defined proportional to the kinetic energy per particle) depends, for the long-range regime $0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$, on $(\ensuremath{\alpha},d,U,N)$ with numerically negligible changes for dimensions $d=1,2,3$, with $U$ the energy per particle and $N$ the number of particles. We verify the finite-size scaling ${T}_{\mathrm{QSS}}\ensuremath{-}{T}_{\ensuremath{\infty}}\ensuremath{\propto}1/{N}^{\ensuremath{\beta}}$ where ${T}_{\ensuremath{\infty}}\ensuremath{\equiv}2U\ensuremath{-}1$ for $U\ensuremath{\lesssim}{U}_{c}$, and $\ensuremath{\beta}$ appears to depend sensibly only on $\ensuremath{\alpha}/d$. Our numerical results indicate that neither the scaling with $N$ of ${T}_{\mathrm{QSS}}$ nor that of ${t}_{\mathrm{QSS}}$ depend on $U$.

DOI
10.1103/physreve.105.044111

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We numerically study, from first principles, the temperature TQSS{T}_{\mathrm{QSS}}TQSS​ and duration tQSS{t}_{\mathrm{QSS}}tQSS​ of the longstanding initial quasi-stationary state of the isolated ddd-dimensional classical inertial \ensuremathα\ensuremath-XY\ensuremath{\alpha}\text{\ensuremath{-}}XY\ensuremathα\ensuremath-XY ferromagnet with two-body interactions decaying as 1/rij\ensuremathα1/{r}_{ij}^{\ensuremath{\alpha}}1/rij\ensuremathα​ (\ensuremathα\ensuremath≥0)(\ensuremath{\alpha}\ensuremath{\ge}0)(\ensuremathα\ensuremath≥0). It is shown that this temperature TQSS{T}_{\mathrm{QSS}}TQSS​ (defined proportional to the kinetic energy per particle) depends, for the long-range regime 0\ensuremath≤\ensuremathα/d\ensuremath≤10\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}10\ensuremath≤\ensuremathα/d\ensuremath≤1, on (\ensuremathα,d,U,N)(\ensuremath{\alpha},d,U,N)(\ensuremathα,d,U,N) with numerically negligible changes for dimensions d=1,2,3d=1,2,3d=1,2,3, with UUU the energy per particle and NNN the number of particles. We verify the finite-size scaling TQSS\ensuremath−T\ensuremath∞\ensuremath∝1/N\ensuremathβ{T}_{\mathrm{QSS}}\ensuremath{-}{T}_{\ensuremath{\infty}}\ensuremath{\propto}1/{N}^{\ensuremath{\beta}}TQSS​\ensuremath−T\ensuremath∞​\ensuremath∝1/N\ensuremathβ where T\ensuremath∞\ensuremath≡2U\ensuremath−1{T}_{\ensuremath{\infty}}\ensuremath{\equiv}2U\ensuremath{-}1T\ensuremath∞​\ensuremath≡2U\ensuremath−1 for U\ensuremath≲UcU\ensuremath{\lesssim}{U}_{c}U\ensuremath≲Uc​, and \ensuremathβ\ensuremath{\beta}\ensuremathβ appears to depend sensibly only on \ensuremathα/d\ensuremath{\alpha}/d\ensuremathα/d. Our numerical results indicate that neither the scaling with NNN of TQSS{T}_{\mathrm{QSS}}TQSS​ nor that of tQSS{t}_{\mathrm{QSS}}tQSS​ depend on UUU.
Abstract We study, by means of first-principle molecular-dynamics, the temperature T QSS of the quasi-stationary state (QSS) of a d -dimensional classical inertial α -Heisenberg ferromagnetic model, consisting of a set of rotators with interactions decaying with distance as r − α ( α ⩾ 0 ) . We find, through detailed finite-size calculations, the scaling relation T QSS − T ∞ ∝ N − β , where T ∞ ≡ U − 1 / 2 , with energy per particle U < U c ( U c = 5 / 6 ), and total number of particles N , which holds for the very long-range interactions regime 0 ⩽ α / d ⩽ 1 , in which the QSS exists. We show that the scaling exponent β does not depend on U but only in the quotient α / d , being β ( α / d ) ≃ 1 for 0 ⩽ α / d ⩽ 0.5 , and decaying linearly for α / d > 0.5 , so <
The duration of the quasistationary states (QSSs) emerging in the ddd-dimensional classical inertial $\ensuremath{\alpha}$-Heisenberg model, i.e., NNN three-dimensional rotators whose interactions decay with distance rij{r}_{ij}rij​ as $1/{r}_{ij}^{\ensuremath{\alpha}}$ ($\ensuremath{\alpha}\ensuremath{\ge}0$), is studied through first-principle molecular dynamics. These QSSs appear for the very-long-range interaction regime ($0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$), for an average energy per rotator U<UcU<{U}_{c}U<Uc​ (Uc=5/6{U}_{c}=5/6Uc​=5/6), and they do not exist for U>UcU>{U}_{c}U>Uc​. They are characterized by a kinetic temperature TQSS{T}_{\text{QSS}}TQSS​, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature TBG>TQSS{T}_{\text{BG}}>{T}_{\text{QSS}}TBG​>TQSS​. We investigate here the behavior of their duration tQSS{t}_{\text{QSS}}tQSS​ when UUU approaches Uc{U}_{c}Uc​ from below, for large values of NNN. The QSS gradually disappears as $U\ensuremath{\rightarrow}{U}_{c}$, while its duration undergoes a critical phenomenon, namely ${t}_{\text{QSS}}\ensuremath{\propto}{({U}_{c}\ensuremath{-}U)}^{\ensuremath{-}\ensuremath{\xi}}$. Universality is found for the critical exponent $\ensuremath{\xi}=1\text{.}67\ifmmode\pm\else\textpm\fi{}0\text{.}02$ throughout the very-long-range interaction regime.
The duration of the quasistationary states (QSSs) emerging in the ddd-dimensional classical inertial $\ensuremath{\alpha}\text{\ensuremath{-}}XY$ model, i.e., NNN planar rotators whose interactions decay with the distance rij{r}_{ij}rij​ as $1/{r}_{ij}^{\ensuremath{\alpha}}$ ($\ensuremath{\alpha}\ensuremath{\ge}0$), is studied through first-principles molecular dynamics. These QSSs appear along the whole long-range interaction regime ($0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$), for an average energy per rotator U<UcU<{U}_{c}U<Uc​ (Uc=3/4{U}_{c}=3/4Uc​=3/4), and they do not exist for U>UcU>{U}_{c}U>Uc​. They are characterized by a kinetic temperature TQSS{T}_{\text{QSS}}TQSS​, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature TBG>TQSS{T}_{\text{BG}}>{T}_{\text{QSS}}TBG​>TQSS​. We investigate here the behavior of their duration tQSS{t}_{\text{QSS}}tQSS​ when UUU approaches Uc{U}_{c}Uc​ from below, for large values of NNN. Contrary to the usual belief that the QSS merely disappears as $U\ensuremath{\rightarrow}{U}_{c}$, we show that its duration goes through a critical phenomenon, namely ${t}_{\text{QSS}}\ensuremath{\propto}{({U}_{c}\ensuremath{-}U)}^{\ensuremath{-}\ensuremath{\xi}}$. Universality is found for the critical exponent $\ensuremath{\xi}\ensuremath{\simeq}5/3$ throughout the whole long-range interaction regime.
An N-sized inertial classical Heisenberg ferromagnet, which consists of a modification of the well-known standard model, where the spins are replaced by classical rotators, is studied in the limit of infinite-range interactions. The usual canonical-ensemble mean-field solution of the inertial classical n-vector ferromagnet (for which n=3 recovers the particular Heisenberg model considered herein) is briefly reviewed, showing the well-known second-order phase transition. This Heisenberg model is studied numerically within the microcanonical ensemble through molecular dynamics. In what concerns the caloric curve, it is shown that, far from criticality, the kinetic temperature obtained at the long-time-limit microcanonical-ensemble simulation recovers well the equilibrium canonical-ensemble estimate, whereas, close to criticality, a discrepancy (presumably due to finite-size effects) is found. The time evolution of the kinetic temperature indicates that a basin of attraction exists for the initial conditions for which the system evolves into a metastable state, whose duration diverges as N--> infinity, before attaining the terminal thermal equilibrium. Such a metastable state is observed for a whole range of energies, which starts right below criticality and extends up to very high energies (in fact, the gap between the kinetic temperatures associated with the metastable and the terminal-equilibrium states is expected to disappear only as one approaches infinite energy). To the best our knowledge, this has never before been observed on similar Hamiltonian models, in a noticeable way, for such a large range of energies. For example, for the XY (n=2) version of the present model, such a behavior was observed only near criticality. It is shown also that the (metastable state) maximum Lyapunov exponent decreases with N like lambda(max) approximately N-kappa, where for the initial conditions employed herein (maximal magnetization), kappa=0.225+/-0.030, both above and below the critical point.
We analyse the low-temperature behaviour of the classical isotropic ferromagnetic Heisenberg model on a two-dimensional square lattice of finite size. Presence of a residual magnetization in a finite-size system enables us to use a low-temperature approximation, which is however more restricting than the usual spin-wave approximation known to give reliable results for the XY model at low temperatures T . For the system considered, we find that the spin–spin correlation function decays as 1/ r η( T ) for large separations r bringing about the presence of a quasi-long-range ordering. We give analytic estimates for the exponent η( T ) in different regimes and support our findings by Monte Carlo simulations of the model on lattices of different sizes at different temperatures.