Thermodynamics of Small Systems

By Peyman Khorsand

In a small system fluctuations can not be ignored, experiments can not be repeated. However, it is still desirable to have some understanding over the non-equilibrium behavior of the system.

Control Parameters vs Fluctuation Variables: For small systems the equation of state and the spectrum of fluctuations are fully determined by so-called control parameters.

The total change in the energy of the system has two components

$dU = \sum_i {\partial U\over \partial x_i}|_{X} dx_i + \sum_a {\partial U\over \partial X_a}|_{x} dX_a$

The first term can be interpretes as internal energy, $dQ$ while the second term can be thought as work $dW$ . For a specific experiment the value of $\Delta Q$ and $\Delta W$ are not reproducible due to the fluctuations in the system, although we can think of their probability distributions.

Probability Distributions: The work and heat probability distributions $P(W)$ and $P(Q)$ characterize the work and heat collected over an infinite number of experiments.

Fluctuation Theorems: The rate at which the system exchange heat with the bath is called the “entropy production“. It is convenient to define $\sigma = Q/Tt$ where $t$ is the interval of time over which the system exchange the heat $Q$ . Associated with the entropy production is a time-dependent probability distribution $P_t(\sigma)$ . Under very general condition the following relation holds (Gallavotti and Cohen) for systems in steady states

$\lim_{t\rightarrow \infty} {k_B \over t} \ln \left( P_{t}(\sigma) \over P_t(-\sigma) \right) =\sigma$

Although the relation holds for infinite limit, it is considered as a good approximation for finite time. Note that the ratio $P_t(\sigma)/P_t(-\sigma)$ for a system in equilibrium with the thermal bath is equal to $1$. This shows that steady state systems are more likely to deliver heat to the bath than it is to absorb heat form bath (Time reversal microscopic laws give birth to non time reversal macroscopic phenomenon, an answer to Loschmidt’s paradox).

The Jarzynski Equality: In a system in contact with thermal bath at temperature T, $\Delta G$ the difference in free energy between equilibrium state $A$ and $B$ with $x_A$ and $x_B$ control parameter.

$\exp \left(- {\delta G \over k_B T}\right)=\langle \exp \left(- {W \over k_B T}\right) \rangle$ ,

the average being taken over infinite number of non-equilibrium processes.

The Crooks Fluctuation Theorem:

Non-Gaussian in the heat pobability distribution:

Generalized JE to arbitrary transition between non-equilibrium steady states:

for more information Ref to “The Nonequilibrium Thermodynamics of Small Systems”, by C. Bustamante, J. Liphardt and F. Ritort

This entry was posted on November 22, 2008 at 5:47 am and is filed under Stochastic Processes. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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Thermodynamics of Small Systems

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