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Indeed, the Entropy Maximization Principle is

fundamentally a Convex Optimization Problem.

The Convex Optimization Problem is guaranteed because the way in

which we've defined the entropy, we do guarantee a global maximum and

this actually found by the Lagrange Multipliers method.

Now, the Principle of Maximum Entropy states that subject to precisely stated

prior data, such as a proposition that expresses testable information.

The probability distribution which best represents

the carnal state of knowledge is the one with maximum entropy.

Now, another way of stating this, take precisely stated prior data or

testable information about a probability distribution function.

Now, consider a set of trial probability distributions that would encode

the prior data that you've collected of those, the one with maximum

information entropy is the proper distribution according to this principle.

The principle was first expounded by Edwin James in two papers in 1957,

where he emphasized a natural

correspondence between Statistical Thermodynamics and Information Theory.

In particular, James offered a new and very general rationale

of why the Gibbesian approach of Statistical Thermodynamics works.

Now, he argued that the entropy of Statistical Thermodynamics is principally

one and the same with the information entropy concept in Information Theory.

Consequently, Statistical Thermodynamics should be seen just as a particular

application of a general tool of logical inference and information theory.

Now, this is a wonderful question.

Let's discuss this with an example.

Now, take all the air in a room and start it out with a special configuration.

The special configuration is where we only occupy a small corner of the room.

In a short amount of time, the molecules will spread out over the room and

occupy the full volume of the room.

Now can the opposite happen?

No.

Because it violates the Second Law of Thermodynamics, right?

The entropy of a confined gas is less than the entropy of the gas that occupies

the entire room.

But, if we time reverse every molecule of the final state,

the air will rush back to the corner of the room.

Now, the problem is, that if we make a tiny error in the motion of just a single

molecule the error grows exponentially with what is known as the exponent.

And instead of going back into the corner of the room they actually go and

fill the entire room.

Now, this is not to say that freak accidents don't happen.

Now, if we wait long enough the air in the room

will accidentally congregate in the corner.

The correct statement is not that unlikely things never happen, but

only that they very rarely happen.

The time that you would have to wait for

the unusual air event to take place is exponential in the number of molecules

Now that's a great question.

The many physical phenomena of interest that involve

quasi-thermodynamic processes that are slightly out of equilibrium.

Let's take some examples.

Now, heat transport by the internal motions in a material,

driven by a temperature imbalance.

Now electric currents, carried by the motion of charges,

in a conductor, driven by a voltage imbalance.

Spontaneous chemical reactions, driven by a decrease in free energy.

Friction, dissipation, quantum decoherence, and so on.

Now all of these processes occur over time with characteristic rates.

And these rates are of crucial importance in engineering.

Now, the field of what is called

Non-equilibrium Statistical Thermodynamics concerns itself with

understanding these non-equilibrium processes at the microscopic level.

Now, Statistical Thermodynamics, the way we have learned it can only

be used to calculate the final result after all these external imbalances

have been removed and the ensemble settles back in equilibrium.

In principle, Non-equilibrium Statistical Thermodynamics could be exact and

ensembles, for instance, for an isolated system could be evolved over time

according to the doministic equations such as the Louisville's Theorem, or

the Quantum Mechanical Version, the Fornierian Equation.

Now, in order to make headway in to modeling this irreversible processes its

necessary to add additional ingredients besides probability and

reversible mechanics.

Now, Non-equilibrium Statistical Thermodynamics is therefore still

an active area of theoretical research as the range

of validity of these additional assumptions continue to be explored.

Now, this is a tricky question.

As we have learned,

there are Thermodynamic Potentials that govern the behavior of specific ensembles.

And these Thermodynamic Potentials are state variables.

Now an important theorem holds for

the state variables that the second order partial derivatives with respect to these

potentials, do not depend on the order in which you perform the derivative.

Let's take an example.

Let's take internal energy U.

Now, we can take a second order derivative of the internal energy U

with respect to the entropy and the volume.

Now we can take this first with respect to volume, then with respect to entropy,

or we can take it with entropy and then with respect to volume.

Now, why is this important?

It turns out that with this,

you can connect the partial derivative of temperature with respect to volume

to the derivative of the pressure with respect to entropy.

Now, these relations are what are known as maximal relations.

Now, the maximal relations are very useful for relating difficult to define

Thermodynamic Quantities to ones that are more easily determined.

In particular, changes in entropy, as you pointed out are difficult to find.

So, it's easy to relate this to changes in pressure, volume or temperature.

Wow, a tricky last question.

Now, any response to the question of what is the meaning of Chemical Potential

is necessarily subjective.

Now what satisfies one person is probably going to be wholly inadequate to

satisfy another.

Now, there are three characterizations of the Chemical Potential

that captures the diverse aspects of its manifold nature.

Now, as a function of position

the Chemical Potential measures the tendency of particles to defuse.

Now, when a reaction may occur.

An of some Thermodynamic Function determines equilibrium.

Now, the Chemical Potential measures the contribution of this change per particle

and for an individual species, the Thermodynamics function's rate of change.

Now, the Chemical Potential also provides a characteristic energy, that is,

the change in energy when one particle is added to the system Holding of course

entropy and the volume constant.

Now, I have to add that these three assertions need to be qualified

by the contextual conditions under which they have been framed.

Now, the first statement captures an essence especially when

the temperature is uniform.

Now, if this is not the case and the temperature varies spatially,

diffusion is somewhat more complex.

Now, the second statement that we described

is valid if the temperature is uniform and fixed.

If instead the total energy is fixed, and

the temperature may vary from place to place then it turns out that

the Chemical Potential divided by the temperature measures this contribution.

Now, when one looks for conditions that describe chemical equilibrium,

one may focus on each locality separately and then,

the division by temperature is inconsequential.

Now, the system's external parameters are the macroscopic environmental parameters.

Such as the external magnetic field or the container volume

that appear in the energy operator, or the energy eigenvalues.

Now, all external parameters are to be held constant

when the derivative in statement three that we described is formed.

The subscript V, that is,

volume, illustrates merely the most common situation.

Note that the pressure does not appear in the Eigenvalues.

So, in the present usage, pressure is not an external parameter.

Thanks a lot for all the questions that you've given us in the office hours.

And I hope you enjoyed the answers.

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Statistical Thermodynamics: Molecules to Machines

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