By Kent W. Mayhew

Blog: Implications to Statistical Thermodynamics

*Implications to Statistical Thermodynamics*

*By Kent W Mayhew*

There are those who firmly believe that the irreversibility of systems can only be explained using statistical thermodynamics. This website and my book are written to show otherwise.

Certainly
when properly used statistical thermodynamics should provide insights to various phenomena. We all can agree that statistical
ensembles can be used to describe the actions of vast numbers of interacting particles i.e. atoms, molecules. And we have the likes
of Boltzmann to thank for this. Even so such ensembles were based upon elastic collisions, and with our improved understanding
that intermolecular collisions are inelastic, means that at most levels how we write the statistical sciences, is in need of a
rethink. Which will be left to others who possess the skills to do so.

A result of believing that irreversibility
can only be explained using statistical arguments i.e. entropy and the second law, has led to claim that the probability construct is
the only way to properly comprehend our universe. This author believes otherwise namely that probabilities give results rather
than not reasons, hence our universe can be better understood using logical constructs. Moreover, if you believe that your language
(no matter what language you speak) is the only language that explains what you witness then you may be in dire
need of a reality check. Remember statistical mathematics is nothing more than a language, albeit a complex eloquent one.

However, issues with statistical thermodynamics go beyond irreversibility and inelastic collisions. And this is one of logic. When
constructive logic is used to formulate a theory then you will be able to move in any direction fom any point in its
structured analysis and at all times arrive at constructive logical results. This is not the case for our traditional understanding
of thermodynamics along with statistical thermodynamics, as we are about to demonstrate!

Implications of our New Understanding to
Statistical Thermodynamics

A fundamental equation to statistical thermodynamics is entropy (*S*) defined in terms of the number
of microstates (*@*) i.e.

*S*=*k*In*@ *
1)

Note: There are other plausible explanations as to why logarithmic functions belong in thermodynamics other than the accepted statistical reasoning.

Arguably eqn 1) is a valid equation because Boltzmann’s constant (k) was designed (or if you prefer
equated) so that the ideal gas law is also valid. That being:

PV=NkT 2)

When contemplating an expanding system, we are often taught the following isobaric isothermal relation:

TdS=dE+PdV 3)

Throughout this website I have discussed that the only logical approach for
dealing with eqn 3) is to consider that dE = system energy change (AKAK Internal Energy), while PdV = Lost work into our atmosphere. Even
so the isothermal entropy (TdS) remains troublesome. Instead compare eqn 3) to what was discussed concerning
the first law, i.e.:

dQin = dEsys + (PdV)atm = dEsys + PatmdVsys 4)

Certainly the RHS of the above two equations
are the same hence eqn 3) is basically a version of the first law whereby one considers that TdS equates to the
energy input dQin.

If you do not understand why eqn 3) lacks clarity while eqn 4) has clarity then see parameters

For
the case of all the energy used to do work onto the surrounding atmosphere, coming from the within the expanding system, then dQin=
0 and eqn 4) becomes:

dE=
-W(lost) = -(PdV)atm 5)

For the
case described by eqn 5) the system's temperature must decline as it does work, in which case one may write:

dEsys =(CvdT)sys =- (PdV)atm 6)

The above is enshrined in constructive logic and is certainly
simple to follow. Now ask: What happens if we insert eqn 1) into eqn 3)

Td*S*=Td(*kIn@) = dE+PdV* 7)

At this point one may ask what is Td(kIn@) in eqn 7)? Seemingly it is now the difference between the system’s energy change and the work done by the system onto its surrounding atmosphere. Perhaps but; what exactly does this have to do with the number of accessible states of either the system, or its surroundings? It an awkward question that is seemingly avoided in traditional thermodynamics! I personally do not have an answer.

A plausible answer is that Td(kIn@) represents the energy input as measured by the increase to the system’s number of accessible states corresponds to the system’s energy increase minus the work done. At first one might say that this holds some merit from a logistics perspective, as herein we are saying that the change to accessible states is energy related. But hang on, equation 7) is suppose to concern the expanding system. Hence: Should not the change in the expanding system’s number of accessible states just be a function of that system’s energy change (total or internal energy)? Again I personnally do not understand the tradiational logic but then again I assume that I am not alone.

Of course the lack of clarity is undeniable in eqn 7). If we rewrite eqn 7) then we obtain:

(TdS)sys= [Td(*kIn@)]sys = dEsys + (PdV*)atm 8)

Now that we have an equation with clarity it now looks like it maybe meaningless

Reconsider what happens if all the energy required comes from the thermal energy within the expanding system as defined by eqn 7). Based on eqn 7), the expanding system's temperature must decrease, so one might actually think that the number of accessible energy states within the expanding system decreases. It becomes an illogical mess if we try to describe what is happening in terms of entropy. Of course remove entropy we return to eqn 6).

Yet traditional
thermodynamics avoids such constructive logic by also claiming for any expanding system that:

TdS = Td(kIn@) = TkIn(V2/V1) 9)

Note: Equation 9) is commonly found in physical chemistry
texts. Now eqn 9) claims that isothermal entropy change is a function of volume change rather than energy per say. Herein the traditional
argument based upon Boltzmann's entropy being related to randomness. But what does randomness or volume have to do with
energy, except for the reality that volume increases of expanding systems signifies work done onto our atmosphere AKAK lost work.

The issue becomes this. Eqn 7) & 9) infer different realities than eqn 9). However one rewrites traditional thermodynamics in terms of a statistical construct must have results that only infer one reality. This is a must if you want it to have any basis on constructive logic. You see traditional thermodynamics is filled with circular logic, just look at the differential shuffle, as inspired by the likes of Gibbs in his famous paper: "Heterogeneity of Homogeneous Substances".

The big circular issue: There are those who claim that Boltzmann’s brilliant statistical mathematics proves that traditional thermodynamics is correct. However we have just seen one of the many disconnects between real thermodynamics and Boltzmann's brilliance and the very basis of all those statistical arguments. Certainly statistical math was built by equating it to known empirically determined results. In other words it was equated to known results and then claimed to prove those very results.

The
above could become the very definition of circular logic! I.e. designing a math and then having a constant (k) in
eqn 1) such that the math equals empirical data is a fundamental to currently taught statistical thermodynamics. And then to claim
that since this math now explains what ewe witness is certainly NOT vested in constructive logic. Those claiming that statistical
thermodynamics proves traditional interpretations to be absolutely correct may just be delusional.

One may
argue that we did not actually deal with statistical thermodynamics in this blog and/or that this author does not fully understand
statistical arguments. And to that I agree as it has been decades since I last studied the subject. But that does not diminish any
of the simple fundamental arguments presented in this blog. The point remains that your complex statistical consideration must match
simpler understandings presented herein, rather than the other way around.

The above does not alter the reality that Boltzmann's math has helped us understand atomic theory.

In another blog I discuss that Boltzmann’s constant (k)
is not a universal constant as previously presumed, rather it is a constant which is only valid here on Earth.

** Conclusions:**

We accept that Boltzmann’s mathematical insights were brilliant. However the number of microstates has nothing to do with volume
or randomness. It would be simpler if it was a relation to the system’s total energy (AKA: Internal energy). Simply put, increase
a systems thermal energy and its number of microstates increases. This will have implications throughout statistical thermodynamics.

Furthermore we must accept how walls influence all forms of thermodynamics.

And finally we might need to accept
the notion that our world might not be governed by probabilities, at least to the extent that this concept has grown throughout the
20^{th} and into the 21^{st} century. This may help the likes of Plank and Mach rest in peace.

Copyright Kent W. Mayhew

The following quote by Authur Eddington demonstrates the purity of human arrogance that can lend itself to the complete indoctrination of a poorly conceived science

“The law that entropy always increases, holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.”

“The law that entropy always increases, holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.”

This website is copyright of Kent W. Mayhew who in 2018 resides in Ottawa Ontario Canada

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