Two-dimensional chaotic thermostat and behavior of a thermalized charge in a weak magnetic field

A two-dimensional version of a chaotic thermostat is investigated. Its structure follows the concept previously introduced by the author [G. J. Morales, Phys. Rev. E 97, 032203 (2018)] to generate a one-dimensional chaotic thermostat, namely, the usual friction force of a deterministic thermostat is supplemented with a self-consistent fluctuating force that depends on the drag coefficient associated with coupling to the heat bath. Azimuthal symmetry requires the thermostat to have two internal degrees of freedom, thus the Martyna-Klein-Tuckerman [G. J. Martyna et al., J. Chem. Phys. 97, 2635 (1992)] model is chosen for the heat bath. The unmagnetized system exhibits two-dimensional diffusive behavior, achieves symmetric Maxwellian velocity distributions in the absence of an external potential, and satisfies the Einstein relation when an external force is applied. The velocity fluctuations display the characteristic exponential frequency spectrum associated with chaotic systems. The model is used to explore the diffusive motion of a thermalized charge in a weak magnetic field and the associated Hall and Pedersen mobilities. Over a range of magnetic field strengths the charge exhibits absolute negative mobility.

Figure 1

Instantaneous values of scaled $y$ and $x$ velocities over a time interval $\tau =10000$ for one realization of the ensemble illustrate the high degree of symmetry resulting from the chaotic fluctuations. Rare and infrequent phase conditions cause the temporary attainment of energies well beyond the thermal energy.

Figure 2

Ensemble-averaged velocity distribution functions in the $y$ and $x$ directions are represented by the different discrete symbols arranged in 41 bins. They are obtained from individual realizations of the type shown in Fig. 1. The red curve is a Maxwellian function with scaled, square average velocity equal to 1. Comparison demonstrates that the velocity distribution functions generated by the chaotic fluctuations are spatially symmetric and closely approximate the thermal equilibrium value.

Figure 3

Autocorrelation functions of the $y$ (red curve) and $x$ (black curve) scaled velocities indicated by the symbol “u,” and of the scaled damping coefficient $\mathrm{\Gamma }$ (blue curve), for a single realization, i.e., not averaged.

Figure 4

Log-linear display of the ensemble-averaged amplitude of the frequency spectrum of the scaled, $x$ velocity, i.e., $\langle \left|{\tilde{u}}_x\left(f\right)\right|\rangle$, for the unmagnetized case, $(w=0)$, is shown as the continuous blue curve. Frequency $f$ is scaled to $t_0^{-1}$. Red straight line is an exponential fit of the form $exp(-2\pi {\tau }_{L}f)$ with ${\tau }_L=0.47$; it indicates that the velocity fluctuations are chaotic.

Figure 5

Scaled $x$ velocity (blue curve), for one realization, is displayed over a small time segment that is arbitrarily chosen. The red curve is a Lorentzian function given by Eq. (18) in which the width is identical to the independently determined width from the slope of the red line that provides an exponential fit to the ensemble-averaged spectrum in Fig. 4. The presence of the Lorentzian pulses shows that the underlying dynamics is chaotic. Similar results, not shown, are found for the $y$ velocity.

Figure 6

Illustration of typical 2D walk for an unmagnetized case, $w=0$ (blue curve), and a magnetized one, $w=1.5$ (red curve). Each point along a curve is the instantaneous, scaled y and $x$ positions for one realization. The time record for both cases corresponds to $\tau =10000$.

Figure 7

Frequency dependence of the ensemble-averaged amplitude of the frequency spectrum of the scaled $x$ position, i.e., $\langle \left|\tilde{x}\left(f\right)\right|\rangle$, is shown as the red curve for an unmagnetized case. Frequency $f$ is scaled to $t_0^{-1}$. A log-log display format is used to illustrate the different behavior occurring on different timescales. The black curve has a functional form of $f^{-1}$, and the blue curve is an exponential function of the form $exp(-2\pi {\tau }_{L}f)$ with ${\tau }_L=0.57$. Similar results, not shown, are found for the $y$ position.

Figure 8

Comparison of the amplitude of the frequency spectrum of the $x$ velocity for a magnetized case $w=1.5$ (red curve) to the unmagnetized case $w=0$ (black curve), using a linear display format in a limited frequency range $f\le 0.5$. Over a larger frequency range the spectrum follows an exponential frequency dependence, as in Fig. 4. The results for the $y$ velocity are identical. The values of the frequencies associated with the A–G peaks are explained in the text. These peaks disappear in the presence of the magnetic field. The unlabeled (red) vertical line corresponds to the scaled cyclotron frequency $f=1.5/2\pi$. The broadening of the cyclotron resonance seen in the red curve is a manifestation of the chaotic fluctuations.

Figure 9

Demonstration of diffusive behavior in the presence of a magnetic field for $w=3$. Blue curve labeled “squared displacement” shows the time dependence of the ensemble average of the squared, scaled displacement $\langle x^2+y^2\rangle {\left(\overline{v}{t}_0\right)}^{-2}$. Red straight lines are linear fits whose slope is proportional to the diffusion coefficient. The deviation from diffusive behavior, i.e., linear increase with scaled time, is caused by the application of a pulsed electric field, indicated at the bottom. Similar results are obtained over the range $w\le 6$.

Figure 10

Magnetic field dependence of effective Langevin drag coefficient obtained from Eq. (24) after calculating ${\mu }_{Ps}$, and ${\mu }_{Hs}$, independently, from the 2D chaotic model by applying a scaled electric field $qE{t}_0/m\overline{v}=0.1$, as illustrated in Fig. 9. Range of scaled cyclotron frequency is $0.05\le w\le 5.5$.

Figure 11

Magnetic field dependence of the scaled diffusion coefficient over the range $0\le w\le 5.5$. Red curve, labeled “2D Chaotic,” is the result of the 2D chaotic model. Blue curve, labeled “Langevin,” is obtained from Eq. (20) with a constant value $\mathrm{\Gamma }=0.5$ that is consistent with the result of the 2D chaotic model for $w=0$. The sharp peaks and steps in the red curve have been ascertained by making small changes in $w$ to resolve them; they are an intrinsic feature of the chaotic dynamics.

Figure 12

Magnetic field dependence of the scaled Hall mobility, i.e., the $xy$ component of the (electrical) mobility tensor. To better establish a comparison with the Langevin model the quantity that is displayed is $w{\mu }_{Hs}$. The red curve, labeled “2D chaotic” is this quantity, calculated from the chaotic thermostat model for a scaled electric field $qE{t}_0/m\overline{v}=0.1$ that points along the $x$ direction, and over a range $0.05\le w\le 5.5$. The blue curve, labeled “$\mathrm{\Gamma }=0.32$,” is the prediction of Eq. (23) for this value, and the black curve, labeled “$\mathrm{\Gamma }=1.73$,” is a similar prediction for this larger value of the scaled drag coefficient.

Figure 13

Magnetic field dependence of the scaled Pedersen mobility, i.e., the $xx$ component of the (electrical) mobility tensor. The red curve, labeled “2D chaotic,” is obtained simultaneously with the results shown in Fig. 12; i.e., a scaled electric field $qE{t}_0/m\overline{v}=0.1$ is applied along the $x$ direction, and the ensemble average velocity along the $x$ direction is calculated over a range of scaled cyclotron frequencies $0.05\le w\le 5.5$. The blue curve, labeled “$\mathrm{\Gamma }=0.5$,” is the prediction of the Langevin model for this value of the scaled drag coefficient which corresponds to the effective drag coefficient associated with the unmagnetized, chaotic mobility. The arrow labeled ANM identifies a limited range of magnetic field values, centered around $w=1.5$, in which the chaotic dynamics results in “absolute negative mobility.”

Figure 14

Threshold behavior of absolute negative mobility (ANM) seen in Fig. 13. Red curve displays the dependence of the scaled Pedersen mobility on the strength of the scaled electric field over a range $0.1\le qE{t}_0/m\overline{v}\le 1$, for a scaled cyclotron frequency $w=1.5$ corresponding to the center of the region of ANM behavior. The phenomenon requires weak electric fields such that $qE{t}_0/m\overline{v}<0.5$. Identical (but asymmetrical) results are obtained when the direction of the electric field is reversed.