Archive for November, 2008

Thermodynamics of Small Systems

November 22, 2008

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

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Foam

November 7, 2008

It is interesting to have a statistical description of foam. In doing so few simple characteristics may come to mind,

Network Properties:

A network can be associated to a foam structure by replacing every cell(bubble) with a node and attach these nodes to their physical neighbors. The statistical properties of this network can be interesting.

Bubble size Distribution:

Various empirical formulae have been proposed by fitting data, including

P(r) \propto {R \over (1+\beta R^2)^4}

P(r) \propto {R^2 e^{-\beta R^2}}

Here r is the radius of the bubble assuming they have spherical shapes and R is the relative size of bubbles to average radius of the population R=r/{\bar {r}}.

Dynamical Aging:

1) Average bubble sizes increase.

2) The probability distribution broadens.

3) Total volume decreases

4) Average life time of a bubble with radius r, we may need to know all the neighbors radii too.

In the case of a single hemispherical bubble at the surface of liquid and gas the bubble burst because of drainage and thinning effect at the top of the buuble. The life time in this case inversely depends on r

\tau \propto {1 \over r}

5) Transition probability of the network after a bubble burst.

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