Abstract
Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker–Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.
Highlights
The second law of thermodynamics introduced the concept of entropy, and states that at the macroscopic scale entropy is nondecreasing for a closed isolated system, i.e., one for which no heat enters or leaves and on which no external work is performed
Mixed in with these new results is a substantial amount of review material, since it makes little sense to present intricacies about differences between Itô and Stratonovich stochastic differential equations or to talk about diffusions on Lie groups to someone familiar only with statistical mechanics or information theory
Note that the necessary and sufficient conditions in (50) for the Boltzmann distribution to be the equilibrium/stationary solution holds even when B and C are dependant on q. This is a generalization of the fluctuation dissipation theorem to the case of complex mechanical systems that can be modeled as a collection of rigid bodies
Summary
The second law of thermodynamics introduced the concept of entropy, and states that at the macroscopic scale entropy is nondecreasing for a closed isolated system, i.e., one for which no heat enters or leaves and on which no external work is performed. The recognition that stochastic mechanical models can be used in place of the original deterministic collision models used to formulate statistical mechanics; The interpretation of these stochastic models as Itô or Stratonovich is irrelevant for systems that have nonzero mass, but that these interpretations provide different results as mass becomes zero—an effect almost never discussed in statistical mechanical works; Conditions for a stochastic mechanical system with configuration-dependent noise and damping to reach equilibrium (i.e., the time-independent probability distribution on phase space) are established, generalizing the Einstein relations and providing a statement that is new and different than those for detailed balance, and represents a new observation in fluctuation-dissipation theory; The aforementioned stationary pdf to which a stochastic mechanical ensemble converges is the Boltzmann distribution from statistical mechanics, which is obtained via a different path than it was in statistical mechanics; Novel solutions to diffusion equations on Lie groups are provided; Inequalities from information theory are extended beyond their original setting (e.g., to include diffusion processes on Lie groups in addition to Euclidean spaces) and used to bound the entropy and rate of entropy production in stochastic mechanical systems such as the rigid Brownian rotor and mobile robots, which are beyond the scope of classical statistical mechanics Mixed in with these new results is a substantial amount of review material, since it makes little sense to present intricacies about differences between Itô and Stratonovich stochastic differential equations or to talk about diffusions on Lie groups to someone familiar only with statistical mechanics or information theory. On a compact space such as the circle or rotation group, the uniform distribution has the absolute maximum entropy of all distributions on those spaces
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