Distributions of Work

Random sampling of pairs of example work distributions that obey the fluctuation theorem.

thermoflow.gaussian_work(key: Array, mu: float, free_energy: float = 0.0, kT: float = 1.0, shape_f: Shape = (1,), shape_r: Shape = (1,)) → Tuple[Array, Array]

Randomly sample Gaussian (normal) distributed work distributions with given mean dissipation.

For Gaussian work distributions that obey the fluctuation theorems, the variance of dissipation is twice the mean, and the mean dissipation is the same forward and reverse.

Parameters:

• key – JAX PRNG key

• mu – Mean dissipation (work minus free energy).

• free_energy – Shift work distributes to obtain this change in free energy.

• kT – Temperature in natural units

• shape_f – Array shape of forward work samples.

• shape_r – Array shape of reverse work samples.

Returns:

(W_f, W_r)

Random samples from forward and reverse work distributions that obey the fluctuation theorem. Units of kT.

Refs:

[1] Nonequilibrium Equality for Free Energy Differences C. Jarzynski Phys. Rev. Lett. 78, 2690 1997

thermoflow.gamma_work(key: Array, a: float, scale: float, free_energy: float = 0.0, kT: float = 1.0, shape_f: Shape = (1,), shape_r: Shape = (1,)) → Tuple[Array, Array]

Randomly sample gamma distributed work distributions with given mean dissipation.

Gamma work distributions occur for the adiabatic compression or expansion of a dilute gas [1]. Parameters corresponding to figure 2 of [1]: (a=3, scale=0.587), (a=9, scale=0.587), (a=6, scale=5.34). For last case a Jarzynski average has large biases, and a bidirectional estimate of free energy is necessary.

Parameters:

• key – JAX PRNG key

• a – Shape parameter of forward gamma work distribution.

• scale – Scale parameter of forward gamma work distribution.

• free_energy – Shift the work distributes to obtain this change in free energy.

• kT – Temperature in natural units

• shape_f – Array shape of forward work samples.

• shape_r – Array shape of reverse work samples.

Returns:

(W_f, W_r)

Random samples from forward and reverse work distributions that obey the fluctuation theorem. Units of kT.

Refs:

[1] Work distribution for the adiabatic compression of a dilute and interacting classical gas, G.E. Crooks, C. Jarzynski. Phys. Rev. E 75 021116 (2007)

thermoflow.exponential_work(key: Array, scale: float, free_energy: float = 0.0, kT: float = 1.0, shape_f: Shape = (1,), shape_r: Shape = (1,)) → Tuple[Array, Array]

Randomly sample exponentially distributed work distributions with given scale. Special case of gamma work distributions.

Parameters:

• key – JAX PRNG key

• scale – Scale of exponential distribution of forward work.

• free_energy – Shift work distributes to obtain this change in free energy.

• kT – Temperature in natural units

• shape_f – Array shape of forward work samples.

• shape_r – Array shape of reverse work samples.

Returns:

(W_f, W_r)

Random samples from forward and reverse work distributions that obey the fluctuation theorem. Units of kT.

thermoflow.two_point_work(key: Array, p: float, free_energy: float = 0.0, kT: float = 1.0, shape_f: Shape = (1,), shape_r: Shape = (1,)) → Tuple[Array, Array]

Sample work from a time symmetric two-point work distribution.

The two-point distribution of work is the time symmetric distribution with the minimum variance for a given mean dissipation [1] [2] [3].

Parameters:

• key – JAX PRNG key

• p – Probability of one point of the distribution.

• free_energy – Shift work distributes to obtain this change in free energy.

• kT – Temperature in natural units

• shape_f – Array shape of forward work samples.

• shape_r – Array shape of reverse work samples.

Returns:

(W_f, W_r)

Random samples from forward and reverse work distributions that obey the fluctuation theorem. Units of kT.

Ref:

[1] https://arxiv.org/abs/1010.2319 [2] https://arxiv.org/abs/1904.07574 [3] https://arxiv.org/abs/2208.11206