Anthony Bartolotta, Sean Carroll, Stefan Leichenauer, and Jason Pollack, 2015
arxiv:1508.02421 [cond-mat.stat-mech]
Abstract: We derive a generalization of the Second Law of Thermodynamics that uses Bayesian updates to explicitly incorporate the effects of a measurement of a system at some point in its evolution. By allowing an experimenter’s knowledge to be updated by the measurement process, this formulation resolves a tension between the fact that the entropy of a statistical system can sometimes fluctuate downward and the information-theoretic idea that knowledge of a stochastically-evolving system degrades over time. The Bayesian Second Law can be written as ΔH(ρm,ρ)+ <Q>F|m≤0, where ΔH(ρm,ρ) is the change in the cross entropy between the original phase-space probability distribution ρ and the measurement-updated distribution ρm, and <Q>F|m is the expectation value of a generalized heat flow out of the system. We also derive refined versions of the Second Law that bound the entropy increase from below by a non-negative number, as well as Bayesian versions of the Jarzynski equality. We demonstrate the formalism using simple analytical and numerical examples.
This web page contains some videos of numerical simulations, pdf illustrations, and Python code associated with the paper “The Bayesian Second Law of Thermodynamics.” In each case, we are simulating the behavior of a simple harmonic oscillator coupled to a heat bath. The images show various versions of the distribution function at various times. Fuller explanation of these images can be found there.
We show three different simulations. In “Static 1,” the oscillator itself is static, and the initial probability distribution is close to thermal equilibrium. In “Static 2,” the oscillator itself is again static, but the initial distribution function is bimodal. In “Drag,” the initial distribution is bimodal, and the minimum-energy value of the oscillator is gradually changed as the system evolves.
First, the Python code:
Next, images of the distributions in pdf:
Finally, the animated videos of the simulation output. Each graph shows the phase space probability distribution with respect to position and momentum at different points in the experiment. Units are chosen such that m = 1, k(t = 0) = 1, and β = 1.
Static protocol, initially near-equilibrium. Evolution of a damped harmonic oscillator coupled to a heat bath in initial thermal equilibrium under a trivial protocol.
Forward evolution:
Reverse evolution, no update:
Reverse evolution, updated after measurement:
Static protocol, initially bimodal. Evolution of a damped harmonic oscillator coupled to a heat bath with known position and magnitude of momentum under a trivial protocol, starting with a distribution that is half large positive momentum and half large negative momentum.
Forward evolution:
Reverse evolution, no update:
Reverse evolution, updated after measurement:
Dragging protocol. Evolution of a damped harmonic oscillator coupled to a heat bath in initial thermal equilibrium under a “dragging” protocol.
Forward evolution:
Reverse evolution, no update:
Reverse evolution, updated after measurement: