Crooks fluctuation theorem

In plain terms

The Jarzynski equality says that a particular average over fast, irreversible runs of a process returns an equilibrium quantity, the free energy difference between the start and the end. Crooks asked a sharper question. Not what the average is, but how likely each individual value of the work is, and how that likelihood compares with the likelihood of the opposite value when the same process is run in reverse.

Picture the process run forwards, from the first equilibrium condition to the second, and separately run backwards, starting in the second equilibrium condition and undoing the driving. In each direction the work done varies from run to run, and one can draw a histogram of the values. Crooks' theorem says that the forward likelihood of doing a certain amount of work, divided by the backward likelihood of recovering that same amount, is the exponential of the difference between that work and the free energy difference, measured in units of the thermal energy. Doing more work than the free energy difference is exponentially more likely forwards than backwards; doing less is exponentially rarer.

Two consequences follow at once. The two histograms cross at exactly one point, the work value equal to the free energy difference, which gives experimenters a direct way to read an equilibrium quantity off two sets of irreversible measurements; it was used to measure the folding free energy of RNA molecules pulled apart and let go with optical tweezers. And the Jarzynski equality falls out in a line, by averaging the exponential of minus the work over the forward histogram and using the fact that the backward histogram, like any histogram, sums to one.

In CSD

In this programme the forward and reverse protocols are the same reading, driving and reading again, with the driving undone in between. The reading probabilities are Born weights, the equilibrium weights are those of the Gibbs state derived from the variational principle, and the driving is a unitary change of the dynamics. The reversed driving has one decisive property: the probability of passing from an energy level of the first Hamiltonian to a level of the second, forwards, equals the probability of passing back, in reverse. This is microscopic reversibility, and it is a consequence of the driving being unitary rather than an added assumption.

With that symmetry in hand, the theorem is a comparison of two equilibrium weights. The forward run weights its starting level by the Gibbs weight of the initial Hamiltonian, the reverse run weights its starting level, which is the forward run's final level, by the Gibbs weight of the final Hamiltonian, and the ratio of the two is the exponential of the work times the ratio of the two partition functions, which is the free energy factor. The corpus states the relation on each pair of readings first, then on the histogram of work values, and derives the Jarzynski equality from the histogram form as a check that the two results are one.

The result says something the second law does not: the irreversibility of a single run has an exact price, and the price is set by how far that run's work sits above the free energy difference. On the constraint surface the reverse protocol is the time reversed Schrodinger flow, itself a Hamiltonian flow that preserves the sector's volume, so both directions sit inside the same volume preserving dynamics. Stating the theorem there, on the two time record law, is the next step of the thermodynamics track.

Mathematically

With the two point measurement setting of the Jarzynski entry (Hermitian H_0 and H_1 on a finite-dimensional space, unitary U, inverse temperature beta), the reverse protocol starts in the Gibbs state of H_1, applies U^*, and reads in the eigenbasis of H_0. Its transition matrix is the transpose of the forward one, because the matrix of U^* between the swapped eigenbases is the adjoint of the matrix of U between the original ones, and squared moduli are invariant under conjugation; its work on the pair (j, i) is minus the forward work on (i, j).

The pairwise theorem is P_F(i, j) Z_0 = P_R(j, i) Z_1 exp(beta W(i, j)), equivalently P_F(i, j) = P_R(j, i) exp(beta (W(i, j) - Delta F)) with Delta F = F_1 - F_0 the difference of the TH3 free energies -log(Z) / beta of the two Gibbs states, and, on pairs of positive transition probability, the ratio form. The proof is the cancellation of the common transition probability and the comparison of the two Gibbs weights exp(-beta E_0(i)) / Z_0 and exp(-beta E_1(j)) / Z_1.

The histogram form defines the work distribution P(w) as the sum of the joint probabilities of the pairs whose work equals w, and states P_F(w) = P_R(-w) exp(beta (w - Delta F)) for every real w. The proof reindexes the reverse pairs along the swap (i, j) to (j, i), under which the reverse work condition -W = -w becomes W = w, and applies the pairwise relation on each surviving term. The work distribution is a probability law on the finite set of work values, and summing exp(-beta w) against it gives the Jarzynski equality in histogram form.

Scope as for the Jarzynski entry: finite dimension, projective energy readings in the spectral theorem's eigenbases, a closed driven system, one temperature at both ends. The reverse protocol uses the same eigenbases, so no choice is hidden in the transpose.

The name

Gavin E. Crooks took his doctorate at the University of California, Berkeley, in 1999, working with David Chandler on nonequilibrium statistical mechanics, and was for many years at the Lawrence Berkeley National Laboratory before moving into quantum computing research in industry.

The fluctuation theorem appeared in two papers: a 1998 article in the Journal of Statistical Physics on free energy differences for microscopically reversible Markovian systems, and the 1999 Physical Review E paper that named the entropy production fluctuation theorem and showed the Jarzynski equality to be its corollary. It was written while Crooks was a graduate student, in the wake of Jarzynski's 1997 equality, and it sharpened that equality from an average into a statement about every value of the work. Delphine Collin, Felix Ritort, Christopher Jarzynski, Steven Smith, Ignacio Tinoco and Carlos Bustamante verified it in 2005 by unfolding and refolding single RNA molecules and reading the folding free energy off the crossing of the two work histograms.

Related entries

• Jarzynski equalityRush a system between two equilibrium conditions as violently as you like, and the average of a particular exponential of the work you did still returns the equilibrium free energy difference exactly. Irreversible experiments measure a reversible quantity.
• Gibbs stateThe state a system settles into at a given temperature is the unique solution to a trade-off between energy and disorder.
• Landauer's principleForgetting costs heat at a rate set by temperature alone, and that's the point at which information stops being an abstraction and acquires a physical price.
• von Neumann entropyA quantum whole can be perfectly known while its parts are as uncertain as they could possibly be, and entropy is where that stops being a metaphor and becomes arithmetic.