Temperature fluctuations in canonical systems: Insights from molecular dynamics simulations

Molecular dynamics simulations of a quasiharmonic solid are conducted to elucidate the meaning of temperature fluctuations in canonical systems and validate a well-known but frequently contested equation predicting the mean square of such fluctuations. The simulations implement two virtual and one physical (natural) thermostat and examine the kinetic, potential, and total energy correlation functions in the time and frequency domains. The results clearly demonstrate the existence of quasiequilibrium states in which the system can be characterized by a well-defined temperature that follows the mentioned fluctuation equation. The emergence of such states is due to the wide separation of time scales between thermal relaxation by phonon scattering and slow energy exchanges with the thermostat. The quasiequilibrium states exist between these two time scales when the system behaves as virtually isolated and equilibrium.

Figure 1

Results of $NVE$ MD simulations. (a) Normalized power spectrum ${\widehat{C}}_{\mathrm{\Delta }K\mathrm{\Delta }K}\left(f\right)$ of kinetic energy fluctuations (filled circles), velocity ACF ${\widehat{C}}_{vv}(f/2)$ (open circles), and phonon density of states $g(f/2)$ (solid line). (b) The kinetic energy ACF $C_{\mathrm{\Delta }K\mathrm{\Delta }K}\left(t\right)$.

Figure 2

Representative fluctuations of the kinetic (blue) and total (orange) energy in the $NVT$ ensemble with the thermostat time constants (a) ${\tau }_r=10$ ps and (b) ${\tau }_r=100$ ps. To enable comparison, the energies were shifted relative to the average values and normalized by the standard deviations. The insets zoom into shorter time intervals to demonstrate the existence of two different time scales of the fluctuations (fast and slow).

Figure 3

Normalized difference between the right- and left-hand sides of fluctuation relations as functions of the averaging time interval $\theta$: Eq. (1) (black solid line), Eq. (2) (red dashed line), and Eq. (3) (blue dotted line). (a) Langevin thermostat with $t_r=10$ ps, (b) Langevin thermostat with $t_r=100$ ps, and (c) natural thermostat.

Figure 4

Normalized power spectra of kinetic and total energy fluctuations in the $NVT$ ensemble with a Langevin thermostat for two different time constants (10 and 100 ps). Square and triangle symbols—kinetic energy; circle and nabla symbols—total energy.

Figure 5

Results of $NVT$ MD simulations with a Langevin thermostat (${\tau }_r=10$ ps). (a) Comparison of the kinetic energy ACF and kinetic-potential energy CCF in the frequency domain. Note that both spectra have the same shape but opposite sign at high frequencies and coincide at low frequencies. (b) Comparison of the total energy ACF and kinetic-total energy CCF in the frequency domain. Both functions show a similar monotonic decrease with frequency and die off above $1/{\tau }_r$.

Figure 6

Energy correlation functions in the time domain obtained by $NVT$ MD simulations with a Langevin thermostat (${\tau }_r=10$ ps). The inset is a zoom into the short-range part of the kinetic energy ACF.

Figure 7

$NVT$ kinetic energy ACF for a Langevin thermostat with ${\tau }_r=10$ ps (red curve) superimposed on the $NVE$ kinetic energy ACF (blue points). The inset shows a zoom into the short-time region.

Figure 8

Anatomy of the “natural” thermostat implemented in this work. (a) Vertical cross section of the simulation block revealing the cubic system under study at the center, the thermostat regions above and below the system, and a fixed shell enclosing both the system and the thermostat. The entire assembly is much longer in the vertical ($z$) direction than shown. Figures (b) and (c) show horizontal ($x-y$) cross sections at the levels indicated by the arrows.

Figure 9

Power spectra of kinetic energy from $NVT$ MD simulations of systems connected to a natural thermostat and two Langevin thermostats with the time constants of 10 and 100 ps.