A Minimal Model for Brownian Vortexes
A Brownian vortex is noise-driven machine that uses thermal fluctuations to extract a steady-state flow of work from a static force field. Its operation is characterized by loops in a probability current whose topology and direction can change with changes in temperature. We present discrete three- and four-state minimal models for Brownian vortexes that can be solved exactly with a master equation formalism. These models elucidate conditions required for flux reversal in Brownian vortexes, and provides insights into their thermodynamic efficiency through the rate of entropy production.
§ I. Introduction
Engines do work by running through a cycle of internal states under the influence of a non-conservative force. Many of the most familiar examples, such as electric motors and internal combustion engines, operate deterministically. Their performance generally is impaired by randomizing influences such as thermal fluctuations, especially as they are miniaturized. Stochastic engines, by contrast, require noise to operate, and do no work at all in the absence of random thermal forcing. Such noise-driven machines have been studied intensively at least since Maxwell's investigations into the foundations of the second law of thermodynamics (1). Most examples are driven out of thermodynamic equilibrium by forces that vary with time, with rocking and flashing ratchets providing particularly popular examples (2). A few others, such as the Feynman (3); (4) and Büttiker-Landauer (5); (6) ratchets function in static force fields, but require the temperature to vary with position or time.
Recently, another class of noise-driven machines has been described that operates in static force fields with conventional heat baths (7); (8). Such systems, known as Brownian vortexes (8), share three defining characteristics
They may be represented as a Brownian particle diffusing in a static force field.
They come to mechanical equilibrium in the absence of thermal fluctuations.
Thermal fluctuations give rise to a steady-state probability flux.
The first characteristic distinguishes Brownian vortexes from thermal ratchets and related stochastic machines that rely on time-dependent forces for their motion (2); (9); (10); (11). The second distinguishes them from deterministic machines such as electric motors. The third requires their probability currents to form closed loops, and thereby gives Brownian vortexes their name. The existence of a steady-state probability flux necessarily implies that the system is driven out of thermodynamic equilibrium, and that the underlying force field does not conserve the particle's mechanical energy. Having assumed that the force field does not vary with time, this implies that it must have a solenoidal component.
The original discussions of Brownian vortexes focused on a particular experimental realization: a single colloidal sphere diffusing in an optical trap (7); (8). The toroidal circulation rolls discovered in this deceptively simple system's probability currents are observed to change direction and topology with continuous changes in the power of the confining laser beam (8). Such observations suggest not only that similar circulation may arise in a wide variety of systems, but also that the phenomenology of such noise-driven machines can be remarkably rich.
This Article offers more general insights into Brownian vortexes by introducing minimal models whose behavior can be computed analytically with master equations. Consideration of these models leads us to distinguish Brownian vortexes into two broad classes: trivial Brownian vortexes that satisfy the three defining characteristics by circulating in one fixed direction, and general Brownian vortexes that dynamically select the number and direction of their circulating rolls. A Brownian pendulum biased by a constant torque is an example of a trivial Brownian vortex. The optically trapped sphere exemplifies the general case.
§ II. Minimal models for Brownian vortexes
A Brownian vortex operates in a static force field,
that features a solenoidal component described by the vector potential . The scalar potential describes a conservative restoring force that brings the system to mechanical equilibrium at zero temperature. The system thus would not move at all in the absence of random thermal forces, and so is distinguished from a deterministic machine. Raising the temperature enables the system to explore the force field around its stable point, with a probability density that we assume to be independent of time. This probability is then advected by the solenoidal component of and relaxes by diffusion to yield the ensemble-averaged steady-state current density,
where is the effective mobility. Conservation of probability, , then requires to form closed loops (8).
Whether or not a particular choice of satisfies the defining conditions for a Brownian vortex is not immediately apparent from this continuum description. In the special case that , the usual Boltzmann solution is retrieved, , and the probability is merely advected by the solenoidal component of the force, . More generally, both and must reflect the influence of . The topology of a circulating steady state and temperature-dependent topological transitions (8) typically would have to be assessed numerically (7); (8). Idealized models admitting analytic solutions are useful, therefore, for elucidating the general characteristics of Brownian vortexes.
§ II.1. Trivial three-state models
The principal characteristics of a Brownian vortex may be abstracted into a biased random walk on a graph such as the example in Fig. 1. This graph's three nodes correspond to locations that are occupied by the particle with probabilities , and the arrows represent transitions among the nodes. The transition that carries the particle from node to node occurs at a rate . Such discrete-state models were introduced by Hill (12) as models for cyclic processes in biology. A similar three-state system has been proposed (13) as a minimal model for Brownian ratchets (14). The minimal Brownian ratchet (MBR) model operates in discrete time steps of duration and, accordingly, describes the transitions between states and in terms of the probabilities per step ; these probabilities are but proxies of the rates . Above and beyond the notational distinction between rates and probabilities, the MBR model (13) requires the system to change states in every time step, a simplifying assumption that imposes the constraint (see also (15)). The discrete-state model for Brownian vortexes instead treats the as non-negative transition rates without additional constraints and so yields the steady-state occupation probabilities
where , and where indexes are computed modulo 3. This solution is appropriately normalized: . The steady-state probability current between any two nodes is then
Minimal models for Brownian vortexes also differ from the MBR model because the transition rates must vary with temperature for to vanish in the low-temperature limit. Two representative models for this temperature dependence reveal general features of the resulting behavior.
§ II.1.1. Kramers model
The defining characteristics of a Brownian vortex can be satisfied, for example, by selecting the non-equilibrium Kramers form for the transition rates:
where is a barrier for transitions arising from the conservative part of the force field, and where is a non-conservative bias that drives the system in the positive direction. Times in the system are scaled by the attempt frequency , and energies are scaled by the thermal energy scale . With these choices, the steady state solution has each site equally populated, , and a temperature-dependent flux
that vanishes in the low-temperature limit provided that . Under these conditions, the three-state model describes stochastic cycle rather than a deterministic machine because it requires thermal fluctuations to operate. In the high-temperature limit, the current is proportional to , which sets the scale of the solenoidal force. The three-state model thus is consistent with the continuum description of Brownian vortexes. Reducing the temperature traps the particle at one of the nodes despite the non-conservative bias. That the flux vanishes at low temperatures completes the identification of the three-state model as a Brownian vortex.
The three-state model also has a deterministic limit, , in which the particle circulates freely around the ring. The circulation rate, however, diminishes with increasing temperature. Like most familiar deterministic machines, therefore, the deterministic three-state model is degraded by thermal noise.
Assuming to be independent of temperature, the entropy production reaches its maximum at temperature
This therefore marks the condition for optimal performance of the Brownian vortex. In the deterministic limit, by contrast, the entropy production rate decreases monotonically with increasing temperature.
§ II.1.2. Advection-diffusion model
An alternative implementation indicated by Fig. 2 assumes that the particle diffuses among the three nodes under the influence of uniform forces that also cause it to drift. To satisfy the first criterion for Brownian vortex operation, the inter-node forces are arranged to bring the particle to mechanical equilibrium on one of the nodes in the low-temperature limit. More specifically, and without loss of generality, we require and to vanish at , so that the particle becomes trapped on node 0. The remaining incoming transitions, and , need not vanish in this limit, and reflect the action of a time-independent force field directing the particle from nodes 1 and 2 toward node 0. Once the fixed point is established, the steady-state probability current at higher temperatures can be controlled through a solenoidal component in the force that is marked by an asymmetry in and , which tends to drive the particle from node 1 to node 2 in Fig. 1. We may assume that the force favors counter-clockwise circulation so that and .
Denoting the magnitude of force by , the transition rates can be written as
where is the particle's mobility and where is the distance between nodes. With these choices, the occupation probabilities and current are
where is work done by the force on a particle as it migrates between nodes. The three nodes are equally populated at high temperature (), and the current approaches .
The occupation probability shifts to node 0 in the low temperature limit and the current vanishes as , independent of the driving force. This model therefore also satisfies the definition of a Brownian vortex.
The entropy production rate,
vanishes as in the high-temperature limit and as in the low-temperature limit. Peak entropy production is achieved at with .
Both realizations of the discrete three-state model describe processes reminiscent of diffusion on a one-dimensional tilted washboard potential, a model nonequilibrium system that has been studied extensively (18); (19); (20). The three-state model's periodicity, however, allows a steady-state to be established (21), which is a defining characteristic of a Brownian vortex. Such systems have been realized experimentally with colloidal spheres moving in holographically structured patterns of light (21); (22); (23). Another example is provided by the biased Brownian pendulum, whose overdamped motions are driven by random thermal forces and a constant torque. All of these models and realizations share in common that the ensemble-averaged steady-state flux always flows in the direction of the applied constant bias. This differs from the motion of a colloidal sphere circulating in an optical tweezer (7); (8), whose probability current changes both direction and topology with changes in the temperature or equivalent control parameters (8). For this reason, we distinguish the more highly constrained three-state model and the systems it describes as trivial Brownian vortexes. Capturing the richer behavior of more general Brownian vortex requires a slightly more elaborate model.
§ II.2. Four-state models of general Brownian vortexes
As the temperature in a Brownian vortex is increased, the diffusing particle can explore more of the force field surrounding its trap. Circulation in its probability current is driven both by the solenoidal component of the force and by diffusion. The sign of the local circulation thus need not be dictated by the sign of the force's local vorticity. Achieving such retrograde circulation is only possible, however, in a force field whose vorticity varies with position. The minimal discrete-state model for a Brownian vortex that undergo flux reversal consequently must have two loops, and thus four states.
Figure 3(a) shows the states and transitions in such a model. We assume that transitions among neighboring nodes are biased by forces, much as in the three-state model. In addition, Fig. 3(a) allows for transitions between nodes 1 and 3, which we assume for simplicity to be unbiased.
The stationary state in this model is a solution of the master equation
subject to continuity,
and conservation of probability
The currents vanish if the system satisfies detailed balance
which is realized when the force field is purely potential and has no solenoidal component. If, on the other hand, the system has no equilibrium because of the solenoidal force, it produces entropy at the rate
The beauty of this model is revealed by the following simple example: suppose that just two transition rates vanish: . In this case, it is obvious that the right branch is not passable, so . Somewhat unexpectedly at first glance, all other currents also stop in the stationary limit, . This happens because site 0 acts like a perfect trap, and the probability gets concentrated on that site.
As for the continuous description in Eq. (1), these transitions are biased by the force field which includes both potential and solenoidal components. The diagram in Fig. 4 shows one particularly simple arrangement of forces that can give rise to a general Brownian vortex with temperature-dependent flux reversal. The forces indicated in the diagram are related to the corresponding energy differences through the length scale introduced by the coarse graining from real continuous system to the discrete one. The forces in Fig. 4 are arranged so that node 0 acts as a trap at zero temperature. In order to describe the temperature-dependent flux reversal observed experimentally (8), the model in Fig. 4 additionally biases the transition from node 3 to node 0 by a factor . With this definition, the transition rates defined in Fig. 3(a) can be written as follows
where , and . Parameters and (or their proxies and ) control the potential and solenoidal parts of the force field. Indeed, if , or , then the condition (16a) is satisfied, while , or , meets the condition (16b). Therefore, as expected, detailed balance is only possible when , or , which corresponds to a purely conservative force field.
where, in this case, and . These denominators are always positive and satisfy .
The results in Eq. (19) allow for the straightforward analysis of the conditions of current reversal. Depending on the parameters and , all three currents can flip their signs, as shown in the diagram in Figure 5.
Currents become very small at high temperature () because the four nodes become nearly equally populated; the remaining small currents
arise because the probability is advected by the solenoidal part of the force, and indeed are proportional to that component of the force. This is consistent with the continuum description in Eq. (2) when the density is independent of position.
The probability currents also vanish in the low-temperature limit (). The analysis of this limit is straightforward, but cumbersome, because the analysis has to be performed separately for every sector of Fig. 5. As an example, we present here the result for the case , or ; in other words, this is the case when force in Fig. 4 is equal in magnitude and opposite in direction to force . In this case, currents are very small because the particle becomes trapped at node 0:
Similar results are obtained if the Kramers' model coupling among the states is replaced by the advection-diffusion model explored in Sec. II.1.2 for the three-state system, although still using the forces depicted in Fig. 4.
§ III. Conclusions
In this Article, we have introduced idealized minimal models for Brownian vortexes, a class of noise-driven machines in which thermal fluctuations are biased into steady-state currents by static force fields. Such systems' departure from equilibrium is mediated by the solenoidal component of their force fields. This observation allows us to distinguish two categories of Brownian vortexes: trivial Brownian vortexes whose currents simply follow the nonconservative solenoidal force, and general Brownian vortexes in which the interplay of advection and diffusion can lead to temperature-dependent flux reversal. The former class is typified by the biased Brownian pendulum and the corrugated optical vortex (13), and the latter by the observed circulation of optical trapped colloidal spheres (7); (8).
All Brownian vortexes share in common that their currents vanish at zero temperature as they lapse into mechanical equilibrium. They are motivated by thermal fluctuations, which enable them to explore their extended force fields. Our minimal models suggest that flux reversal is intimately connected to the possibility for topological transitions in the induced current density, which in turn depends on properties of the force field. How this works is clear for the discrete minimal models we have studied. The conditions under which flux reversal occurs in continuous systems remain to be elucidated.
Of particular interest is the observation that the rate of entropy production varies smoothly during flux reversals in the minimal model for general Brownian vortexes. Although such transitions are continuous in our minimal model, it is conceivable that temperature-dependent transport transitions in more elaborate systems could be discontinuous. Such abrupt transitions might be associated with pattern formation in extended systems undergoing Brownian vortex circulation.
The master equation formalism used to study discrete-state Brownian vortex models also provides a useful abstraction with which to seek Brownian vortex behavior in other systems. These idealized minimal models are examples of biased random walks on graphs, which have been studied extensively in such fields as biology (12) and economics (24). Previous studies in such fields have focused on the role of time-dependent processes in driving non-equilibrium behavior. The present work demonstrates that static solenoidal forces can play a complementary role in generating cycles. In the chemical or biological context, solenoidal driving might be provided by an external energy source that maintains constant concentrations of reagents or products. In finance, it might arise from networks of contractual obligations. Stochastic cycles emerging in these systems may thus be examples of Brownian vortexes, and their behavior more fully explored through this relationship to mechanical models.
It also will be interesting to realize the models described here directly in experiments. For instance, a protein complex or chemical system with reaction pathways described by Kramers' model driven out of equilibrium by a non-conservative influence of any nature may profitably be described in the language of Brownian vortexes, particularly with regard to its non-trivial temperature dependence. Brownian vortexes based on drift and diffusion among discrete nodes also can be constructed using optical tweezers or with diode-based electronics. We will leave these topics for further studies.
This work was supported by the National Science Foundation through Grant number DMR-0855741. B.S. acknowledges support from the Kessler Family Foundation. D.G.G. acknowledges the support of a Joliot-Curie Visiting Professorship and the hospitality of the École Supérieure de Physique et de Chimie Industrielles during part of this study.
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