Energy-Efficient Pseudo-Ratchet for Brownian Computers through One-Dimensional Quantum Brownian Motion

TL;DR

The paper introduces a one-dimensional quantum Brownian motion model in which momentum-space is partitioned into disjoint subspaces by a resonance condition, enabling intrinsic unidirectional transport without external forcing and without net energy dissipation. The dynamics reduce to an advection-diffusion equation with momentum-dependent coefficients $\sigma(P)$ and $D(P)$, and unidirectional transport arises at local equilibrium within each subspace, while quantum dissipation persists only in the quantum regime. By preparing a nonfactorizable Gaussian initial state that creates a coordinate–momentum correlation, the authors demonstrate transient spatial contraction with an apparent negative diffusion, yet the total entropy respects the H-theorem due to a reduction in mutual information $S_I^{X:P}$ between $X$ and $P$. This framework suggests a route to ultra-low-energy Brownian computing, where transport direction and spatial concentration can be controlled via intrinsic quantum correlations rather than external work, with entropy accounting ensuring thermodynamic consistency.

Abstract

Brownian computers utilize thermal fluctuations as a resource for computation and hold promise for achieving ultra-low-energy computations. However, the lack of a statistical direction in Brownian motion necessitates the incorporation of ratchets that facilitate the speeding up and completion of computations in Brownian computers. To make the ratchet mechanism work effectively, an external field is required to overcome thermal fluctuations, which has the drawback of increasing energy consumption. As a remedy for this drawback, we introduce a new approach based on one-dimensional (1D) quantum Brownian motion, which exhibits intrinsic unidirectional transport even in the absence of external forces or asymmetric potential gradients, thereby functioning as an effective pseudo-ratchet. Specifically, we exploit that quantum resonance effects in 1D systems divide the momentum space of particles into subspaces. These subspaces have no momentum inversion symmetry, resulting in the natural emergence of unidirectional flow. We analyze this pseudo-ratchet mechanism without energy dissipation from an entropic perspective and show that it remains consistent with the second law of thermodynamics.

Figures (7)

• Figure 1: Unidirectional transport of a one-dimensional quantum Brownian particle under the parameters chosen as $D^\ast/D_{\rm u}=1$. All values are depicted in the units defined in Sec. \ref{['Model and Kinetic equation']}. The temperature is set to $T=1$. Unlike free particle propagation, even when a particle is initially given negative momentum, the particle propagation at local equilibrium $t\gtrsim\tau_{\mathrm{rel}}$ occurs in the positive spatial direction. Note that during the relaxation period ($t=0$ to $t=\tau_{\mathrm{rel}}$), the momentum distribution of the particle relaxes to the thermal equilibrium in each subspace with the temperature of the phonons. (a) Initial Wigner distribution at $t=0$ with a momentum peak at $P^{\prime}=-3.1$ with width $\Delta X = 3$. This peak momentum belongs to the subspace $P_0=0.9$, which means $P^{\prime}=P_2(0.9)=-3.1$. (b) Wigner distribution immediately after reaching local equilibrium at $t=\tau_{\mathrm{rel}}$. Since momentum transfer occurs only within the subspace to which the initial momentum belongs, the equilibrium momentum distribution has a peak at $P_0=0.9$ with a small side peak at $P_2(0.9)=-3.1$. As the temperature increases, additional side peaks emerge at discrete momenta $P_{\nu}(0.9)$. (c) and (d) are the Wigner distribution at $t=\tau_{\mathrm{rel}}+50$ and $t=\tau_{\mathrm{rel}}+100$, respectively. The distribution moves in the positive $X$ direction with the sound velocity $\sigma(0.9)>0$.
• Figure 2: Apparent negative diffusion with a single nonfactorizable Gaussian wave packet under the parameters chosen as $D^\ast/D_{\mathrm u}=1$. All values are depicted in the units as described in Sec. \ref{['Model and Kinetic equation']}. The left side shows bird's-eye views of the Wigner distribution function, and the right side presents contour plots. The arrows in the contour plots indicate the direction of momentum-dependent advection. The temperature is set to $T=1$ in the aforementioned units. (a) Initial distribution at $t=0$ with the negative shift parameter $\alpha=-40$ (in units of $x_{\mathrm u}/p_{\mathrm u}$). The peak of the Gaussian is set at the origin $(X^{\prime}, P^{\prime})=(0,0)$ with its width $\widetilde{\Delta X} = 3$. (b) Wigner distribution immediately after reaching local equilibrium at $t=\tau_\mathrm{rel}$. Since the initial momentum peak is set at $P_0=P^{\prime}=0$, side peaks appear at $P_1(P_0)=2$ and $P_{-1}(P_0)=-2$. (c) Wigner distribution at $t=\tau_\mathrm{rel}+15$. Although diffusion spreads the wave packet, advection dominates, leading to the contraction of the wave packet in the spatial direction. (d) Wigner distribution at $t=\tau_\mathrm{rel}+30$. The tilt in the contour of the distribution has almost vanished due to advection. (e) Wigner distribution at $t=\tau_\mathrm{rel}+40$. At this stage, both the diffusion and the advection contribute to the spreading of the wave packet.
• Figure 3: Time evolution of the phenomenological diffusion coefficient $D^{(x)}(t)$ and the effect excluding the diffusion effect $A^{(x)}(t) = D^{(x)}(t) - \bar{D}$ in the situation depicted in Fig. \ref{['apparently_negative_diffusion']}. The relaxation time (in units of $t_{\mathrm u}$) is $\tau_\mathrm{rel} \approx 1$ under the parameters chosen as $D^\ast=D_{\mathrm u}$. The cross marks represent $D^{(x)}(0)$, the solid line shows $D^{(x)}(t \gtrsim \tau_\mathrm{rel})$, the triangle marks indicate $A^{(x)}(0)$, and the dashed-dotted line depicts $A^{(x)}(t \gtrsim \tau_\mathrm{rel})$. The time $t_d$ satisfies $D^{(x)}(t_d) = 0$, which corresponds to the threshold time, at which contraction turns into expansion, resulting in the minimum width of the spatial distribution. Additionally, the time $t_a$ satisfies $A^{(x)}(t_a) = 0$, at which the negative tilt in the contour of the distribution turns into a positive one.
• Figure 4: The time evolution of entropies in the situation discussed in Fig. \ref{['apparently_negative_diffusion']} is shown at the initial time $t=0$ and in the region $t\gtrsim\tau_\mathrm{rel}$. The black circle represents $S(0)$, the solid line represents $S(t\gtrsim \tau_\mathrm{rel})$, the red cross represents $S^X(0)$, the dashed line represents $S^X(t\gtrsim \tau_\mathrm{rel})$, the blue square represents $S^P(0)$, the dotted line represents $S^P(t\gtrsim \tau_\mathrm{rel})$, and the green triangle represents $S_I^{X:P}(0)$, with the dot-dashed line representing $S_I^{X:P}(t\gtrsim \tau_\mathrm{rel})$. Note that although the black solid line and blue dotted line appear to show a discontinuous jump between $t=0$ and $t=\tau_{\mathrm{rel}}$, these values are actually continuous. This apparent discontinuity arises because the momentum relaxation process occurs on a much shorter time scale than that of spatial relaxation. We do not plot the entropy evolution during this rapid relaxation period, as no analytical expression is available in this interval. The time at which the entropy of the spatial distribution $S^X(t)$ reaches its minimum is $t_X\sim 15.72$ (in units of $t_{\mathrm u}$), and the time at which the mutual information $S_I^{X:P}(t)$ reaches its minimum is $t_I\sim 29.04$. These times coincide with $t_d$ and $t_a$ in Fig. \ref{['fig_phenomenological_diffusion_constant']}.
• Figure 5: Discrete momentum transitions in the 1D quantum system and the resulting asymmetric Maxwell distribution. All units are as described in the model introduction. (a) Discrete momentum subset accessible from $P_0=0.7$. (b) Momentum equilibrium distribution in the subspace ${\cal S}_{P_0=0.7}$ at $T=1$ in the temperature unit $T_{\mathrm u}$. Only a few momentum states near the origin are thermally excited. (c) Momentum equilibrium distribution in the same subspace at $T=5$. A broader range of momentum states is excited.