Investigation of a chaotic thermostat

A numerical study is presented of a free particle interacting with a deterministic thermostat in which the usual friction force is supplemented with a fluctuating force that depends on the self-consistent damping coefficient associated with coupling to the heat bath. It is found that this addition results in a chaotic environment in which a particle self-heats from rest and moves in positive and negative directions, exhibiting a characteristic diffusive behavior. The frequency power spectrum of the dynamical quantities displays the exponential frequency dependence ubiquitous to chaotic dynamics. The velocity distribution function approximates a Maxwellian distribution, but it does show departures from perfect thermal equilibrium, while the distribution function for the damping coefficient shows a closer fit. The behavior for the classic Nosé-Hoover (NH) thermostat is compared to that of the enlarged Martyna-Klein-Tuckerman (MKT) model. Over a narrow amplitude range, the application of a constant external force results quantitatively in the Einstein relation for the NH thermostat, and for the MKT model it differs by a factor of 2.

Figure 1

Chaotic phase space $(\mathrm{\Gamma },u)$ generated when the fluctuating force and damping force act together for the MKT model (left panel) and NH model (right panel) for a particle initially at rest. Display shows points sampled after 100 computational steps over the time interval $0<\tau <5000$.

Figure 2

Typical time dependence of the self-consistent, fluctuating force for (a) the MKT model and (b) the NH model. Note the small time interval shown, $0<\tau <100$, in comparison to that in Fig. 1.

Figure 3

Effect of fluctuating force on the velocity autocorrelation function for a particle with an initial scaled velocity $u=0.5$ for (a) MKT model and (b) NH model. The top blue curve in each panel only includes the frictional force. The bottom red curves show the decay of the correlation when the fluctuating force is included.

Figure 4

Small-scale temporal structure of the scaled velocity in the NH model. (a) The interval, $600<\tau <700$, during which several sharp pulses are present. Superposed on the black time trace is a red curve consisting of a Lorentzian function chosen to line up with one pulse. (b) An expanded view. The width of the Lorentzian is ${\tau }_L=0.51$. Similar results are obtained for the other narrow pulses.

Figure 5

Phase space ($\mathrm{\Gamma },u$) evolution associated with Fig. 4. The blue curve corresponds to the longer-time interval shown in the top panel, Fig. 4. The red curve is the phase-space trajectory associated with the expanded pulse in the bottom panel, Fig. 4. The flight around the quasistationary point ($\mathrm{\Gamma }=0,u=1$) is the Lorentzian pulse, lasting a characteristic time ${\tau }_L=0.51$.

Figure 6

Phase-averaged, frequency spectrum of velocity variable for the NH model. (a) A log-linear display and (b) a log-log display. The blue curve is the numerical result, averaged over 200 initial values of $\theta$. The red curves correspond to an exponential frequency dependence with a Lorentzian temporal width ${\tau }_L=0.54$.

Figure 7

Coherent peaks in the phase-averaged, frequency spectrum of velocity variable for the NH model. To be compared to Fig. 6. The red frequency markers labeled (a), (b), and (c) correspond to the intrinsic frequencies of the system: $a=1.56$, $b=\sqrt{2}$, $c=0.59$. The nonlinearly generated, spectral lines are all combinations of these basic frequencies, as identified by the arrows next to them.

Figure 8

The velocity distribution functions for the MKT and NH models are shown in (a), and the corresponding distributions for the $\mathrm{\Gamma }$ damping coefficient are displayed in (b), all using a linear scale. The MKT curves are smoother because they are averaged over an ensemble that is 50 times larger than for the NH system.

Figure 9

Distribution functions for the MKT model displayed in a log-linear scale. The blue crosses are the individual values from the numerical solution and the red curve is a Maxwellian distribution with ${\overline{u}}^2=1$. (a) Velocity distribution function. (b) Distribution function of $\mathrm{\Gamma }$ damping coefficient.

Figure 10

Time evolution of the averaged, square displacement, $〈{(▵x/l)}^2〉$, for the MKT and NH models demonstrates that the behavior is diffusive. The blue straight lines are fits to the numerical solution; their slope determines the value (twice) of the respective diffusion coefficient. The MKT curve shows smaller fluctuations because it is averaged over an ensemble that is 50 times larger than for the NH system.

Figure 11

Response to a pulsed, constant force for the NH model. Red curve is the scaled, average-velocity, ${\langle u\rangle }_{\tau ,\theta }$, divided by the peak strength of the force, i.e., the mobility. The average is over time and over initial phases. A steady flow is established from which the Einstein relation is tested, using the diffusion coefficient from Fig. 10. The temporal shape of the applied pulse is shown near the bottom. After the pulse is off, the mean velocity decays.

Figure 12

Transition from diffusive to ballistic behavior for the NH model. The red curve is the temporal evolution of the averaged, square displacement, $〈{(▵x/l)}^2〉$, resulting from the application of a pulsed force. The black curve is the diffusive behavior without the force. The two blue, straight lines signal path of diffusive behavior obtained from Fig. 10. They are connected by a regime of ballistic behavior (parabolic) during application of the pulse. After the pulse is off, the diffusive behavior returns.