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paweld
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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: [tex] (\Delta S)_{U,V,N_1,N_2,...,N_k} > 0 [/tex]
(where [tex] (\Delta S)_{U,V,N_1,N_2,...,N_k} [/tex] 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
[tex] U,V,N_1,N_2,...,N_k [/tex]).
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:
[tex] (\Delta U)_{S,V,N_1,N_2,...,N_k} < 0 [/tex]
(where [tex] (\Delta U)_{S,V,N_1,N_2,...,N_k} [/tex] 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
[tex] S,V,N_1,N_2,...,N_k [/tex]).
Please feel free to share your opinion about these formulations.
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: [tex] (\Delta S)_{U,V,N_1,N_2,...,N_k} > 0 [/tex]
(where [tex] (\Delta S)_{U,V,N_1,N_2,...,N_k} [/tex] 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
[tex] U,V,N_1,N_2,...,N_k [/tex]).
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:
[tex] (\Delta U)_{S,V,N_1,N_2,...,N_k} < 0 [/tex]
(where [tex] (\Delta U)_{S,V,N_1,N_2,...,N_k} [/tex] 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
[tex] S,V,N_1,N_2,...,N_k [/tex]).
Please feel free to share your opinion about these formulations.
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