Maximum entropy and minimum energy principles.

[paweld]

I'm looking for precise formulation of maximum entropy (second law
of thermodynamics) and minimum energy principles.

For example the following formulation of the second law (maximum entropy principle)
which can be found in Callen's Thermodynamic book is quite good:

There exists a function (called the entropy S) of the extensive
parameters of any composite system, defined for all equilibrium states and
having the following property: The values assumed by the extensive parame-
parameters in the absence of an internal constraint are those that maximize the
entropy over the manifold of constrained equilibrium states. (I don't know
if according to these definition the system has to be isolated or only
extensive parameters of the whole system have to be held constant;
by constraint it is meant any separation between susbsytems of
the system e.g. some diathermic/ adiabatic/ semipermable ... walls).

In my opinion the above formulation is better then the more popular one which
does not require usage of constraints (e.g. In an isolated system the entropy
tends to a maximum at constant energy), because if an isolated system is
not in equilibrium, we can associate no entropy with it, and if it is in equilibrium,
its entropy can no longer increase. So the most maximum entropy princile is
quite useless because we cannot use it to determine an equillibrium state
of our system. The possible way to resolve this problem is to consider the composite
system (system consisted of some subsystems separated by partitions, i.e. system
with constraints) as e.g. Callen does (we are considering then only equillibrium states
and only change form more restrictive equillibrium states to less restricitive).
The problem is that I'm not certain what types of constraints are
admissible (can we remove constraints if it require work or heat transfer from
outside but the total inner energy of the system does not change?).

I wonder if the following formulation of second law (which is slight modification
of Callen version) is correct:

There exists a function (called the entropy S) of the extensive
parameters of any composite system, defined for all equilibrium states and
having the following property: $$(\Delta S)_{U,V,N_1,N_2,...,N_k} > 0$$
(where $$(\Delta S)_{U,V,N_1,N_2,...,N_k}$$ is the difference of entropy between
equillibrium state without any constraints and any equillibrium state with some
constraints which can be dismantled without change of extensive parameters
$$U,V,N_1,N_2,...,N_k$$).

If the above definition is correct it's quite simple to prove the principle of minimum
energy and its formulation is "symmetric" to the formulation of principle of maximum
entropy, namely:

$$(\Delta U)_{S,V,N_1,N_2,...,N_k} < 0$$
(where $$(\Delta U)_{S,V,N_1,N_2,...,N_k}$$ is the difference of inner energy
between equillibrium state without any constraints and any equillibrium state with some
constraints which can be dismantled without change of extensive parameters
$$S,V,N_1,N_2,...,N_k$$).

Please feel free to share your opinion about these formulations.

 

[eljose79]

Thank you for your post and for sharing your thoughts on the formulation of the second law of thermodynamics and the principles of maximum entropy and minimum energy.

I can provide you with a more precise formulation of the maximum entropy principle and the minimum energy principle. The second law of thermodynamics states that the entropy of an isolated system will always increase over time, until it reaches a maximum value at equilibrium. This can be mathematically expressed as:

dS ≥ 0

where dS represents the change in entropy, and the greater than or equal to sign indicates that the entropy can either remain constant or increase, but it can never decrease.

The principle of maximum entropy states that the equilibrium state of a system will be the one with the maximum entropy, given the constraints and the available information about the system. This can be expressed as:

S = max

where S is the entropy of the system, and the max symbol indicates that the entropy is at its maximum value.

On the other hand, the minimum energy principle states that a system will always tend towards the state with the minimum energy, given the constraints and available information. This can be expressed as:

U = min

where U is the internal energy of the system, and the min symbol indicates that the energy is at its minimum value.

I hope this helps clarify the formulations of the second law and the principles of maximum entropy and minimum energy. However, I must note that these principles only hold true for isolated systems. In open systems, energy and entropy can be exchanged with the surroundings, and the principles may not apply. Additionally, the constraints in a system can affect the values of entropy and energy, and it is important to consider all constraints when applying these principles.

Thank you for your interest in thermodynamics and for your contribution to the discussion on these important principles.

Scientist