The stochastic discrete nonlinear Schrödinger equation: microscopic derivation and finite-temperature phase transition

TL;DR

The paper derives a stochastic discrete nonlinear Schrödinger equation (SDNSE) by coupling the one-dimensional DNSE to a microscopic heat bath, yielding a model that obeys detailed balance and relaxes to the constrained canonical distribution $\rho_β(\{c_k,\bar{c}_k\})=Z_β^{-1} e^{-β H_S} δ(N-1)$ with $N=1$. It analyzes finite-temperature behavior, identifying a first-order phase transition between a disordered background and a localized breather phase controlled by the rescaled temperature $\bar{β}=αβ/(2L)$, supported by a caloric equation of state and a mean-field calculation that places the critical value at $\bar{β}_c ≈ 2.46$. The transition exhibits phase coexistence and metastability in finite systems, and the study reveals stochastic-resonance-like effects of the bath noise on breather formation. By linking positive-temperature stochastic dynamics to negative-temperature DNSE physics, the work suggests experimental pathways to realize finite-temperature breather transitions in open, bath-coupled setups.

Abstract

We study a stochastic version of the one-dimensional discrete nonlinear Schr{ö}dinger equation (DNSE), which is derived from first principles, and thus possesses all the properties required by statistical mechanics, such as detailed balance and the H-theorem. The stochastic version shows disordered and localised dynamics, and displays a corresponding phase transition at a finite temperature value. The phase transition can be captured in a quantitative way by a mean-field type approach. The corresponding coarsening dynamics shows an unexpected dependence on the noise strength, which is reminiscent of stochastic resonance. The phase transition is linked with negative temperature phase transitions, which have been reported recently for the Hamiltonian dynamics of the DNSE. Our approach gives a clue to how these negative temperature phase transitions can be implemented in experimental setups, which are inevitably coupled to a positive temperature heat bath.

Figures (9)

• Figure 1: (a) Time traces of the energy $E(t)-2$ for parameter values $(\alpha,\beta)$ with positive $\alpha$ and of $2-E(t)$ for conjugate parameter settings $(-\alpha,-\beta)$ obtained from numerical simulations of eq. \ref{['2.2.1']} with system size $L=64$, noise strength $\sigma=0.3$ and stepsize $\tau=0.01$, generated from a localised initial condition $c_n(0)=\delta_{n,32}$ by skipping a transient of length about $\Delta t = 1.6 \times 10^5$ to ensure stationary behaviour. Conjugate parameter pairs: $(\alpha=16,\beta=10)$ (green), $(\alpha=-16, \beta=-10)$ (pink) and $(\alpha=-16,\beta=10)$ (blue), $(\alpha=16,\beta=-10)$ (orange). (b) The associated energy autocorrelation functions on a semi-logarithmic scale obtained from a time series of length about $T=2.6 \times 10^{6}$, showing again the coincidence of data with conjugate parameter values. Parameter settings and colour coding as in part (a).
• Figure 2: (a) Time traces of the energy $E(t)$ for a system of size $L=64$ with parameter values $\alpha=16$, $\beta=10$ and different noise strength: $\sigma=0.01$ (green), $\sigma=0.03$ (orange), $\sigma=0.1$ (blue) and $\sigma = 0.3$ (pink). Data have been obtained from a simulation of eq. \ref{['2.2.1']} with stepsize $\tau=0.01$ and initial condition $c_n(0)=\delta_{n,32}$. The energy is shown as a function of the rescaled time variable $\sigma^2 t$. The data collapse shows that $\sigma^2 t$ sets the relevant time scale of the system. (b) Autocorrelation function of the energy in the stationary state, as a function of $\sigma^2 t$ for the parameter setting (and colour coding) of part (a). Data have been computed from a time series of length about $T=2.6 \times 10^{6}$ by skipping a transient of length about $\Delta t=16 \times 10^4$. The deviations from exponential decay for small values of $\sigma$ are due to sampling errors in the numerical computation.
• Figure 3: (a) Mean energy $E(\beta)$ computed from time traces of eq. \ref{['2.2.1']} of length about $T=2.6 \times 10^{6}$ (with longer time traces $T=10^{7}$ in the transition region) skipping a transient of length about $\Delta t= 1.6 \times 10^{5}$ for system size $L=64$, noise strength $\sigma=0.3$, and two different values of $\alpha$: $\alpha=8$ (green) and $\alpha=16$ (orange). Computations have been performed by an adiabatic parameter upsweep (open symbols) and downsweep (full symbols) of $\beta$, respectively. The black dashed line is the analytical result eq. \ref{['e.16']} obtained for the case $\alpha=0$. (b) Variance of the energy for the same parameter setup (and colour coding) as in part (a).
• Figure 4: (a) Variance of the energy in dependence on the rescaled temperature variable $\bar{\beta}=\alpha \beta/(2 L)$ for $\sigma=0.3$ and $L=32$ (open symbols), $L=64$ (full symbols), $\alpha=8$ (green), $\alpha=16$ (orange), $\alpha=32$ (blue), showing that the location of the phase transition depends essentially on the parameter $\bar{\beta}=\beta\alpha/2L$. Data have been computed by adiabatic parameter upsweeps (solid) and downsweeps (dashed) taking time averages over an interval of length about $T= 2.6 \times 10^{6}$ while skipping a transient of length about $\Delta t = 1.6 \times 10^{5}$. The three black points indicate the particular temperature values used in Figure \ref{['fig33']}. (b) Mean energy $E(\bar{\beta})$ for a system of size $L=128$ (green) and $L=64$ (orange) with $\alpha=16$ and $\sigma=0.3$, showing hysteresis due to extensive equilibration times for larger system sizes. Data have been computed by adiabatic parameter upsweeps (full symbols, solid) and downsweeps (open symbols, dashed) taking time averages over an interval of length about $T=2.6 \times 10^{6}$ while skipping a transient of length about $\Delta t=1.6 \times 10^{4}$. The black dash-dotted line is the equilibrium energy of the system predicted by equation \ref{['4.1a']}, with the blue dashed line indicating analytic estimates of metastable states (see section \ref{['sec:4']} for details).
• Figure 5: Space-time density plots of the pattern of amplitudes, $|c_n(t)|^2$, in the stationary state as obtained from numerical simulations of eq. \ref{['2.2.1']} with stepsize $\tau=0.01$ for a system of size $L=64$, noise strength $\sigma=0.3$ and $\alpha=16$. Temperature values: (a) $\bar{\beta}_{low} = 2.175\quad(\beta_{low}=17.4)$, (b) $\bar{\beta}_{crit} = 2.4 \quad(\beta_{crit}=19.2)$, and (c) $\bar{\beta}_{high} = 2.5 \quad(\beta_{high}=20)$(see as well Figure \ref{['fig33']}(a)), Colour coding: from blue ($|c|^2=0.01$) to yellow ($|c|^2=0.99$). Panels (d), (e), and (f) show the corresponding time evolution of the total energy $E(t)$. Data have been obtained using a localised initial condition $c_n(0)=\delta_{n,32}$, skipping a transient of length about $\Delta t=1.6 \times 10^{5}$ to ensure stationary behaviour.