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I am reading Callen's Thermodynamics and an introduction to Thermostatics.

Initially the author describes the temporal nature of macroscopic measurements as-

A macroscopic observation cannot respond to those myriads of atomic coordinates which vary in time with typical atomic periods. Only those few particular combinations of atomic coordinates that are essentially time independent are macroscopically observable.
The word essentially is an important qualification. In fact we are able to observe macroscopic processes that are almost, but not quite, time independent. With modest difficulty we might observe processes with time scales on the order of $10^{-7}$s or less. Such observable processes are still enormously slow relative to the atomic scale of $10^{-15}$s. It is rational then to first consider the limiting case and to erect a theory of time-independent phenomena. Such a theory is thermodynamics.

Thermodynamics, in contrast, is concerned with the macroscopic consequences of the myriads of atomic coordinates that, by virtue of the coarseness of macroscopic observations, do not appear explicitly in a macroscopic description of a system.

After reading this I get very confused. I have doubt in this abstract definition of macroscopic variables from a long time.

Author said that macroscopic variables are variables that it acquire some constancy in both time and spatial averaging.

  1. Why are we concerned with atomic time scale? As thermodynamical phenomenon like expansion of gas etc occurs at normal time scale?

  2. Also what the author means by saying that macroscopic variable will become constant in time and spatial averaging?

I have only studied Thermodynamics. I know a very little about statistical mechanics.

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1 Answer 1

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I am going to take remarks from my answers at https://physics.stackexchange.com/a/340874/59023 and https://physics.stackexchange.com/a/620654/59023 to address this question.

Why are we concerned with atomic time scale? As thermodynamical phenomenon like expansion of gas etc occurs at normal time scale?

The question is whether we need to concern ourself with atomic processes when dealing with macroscopic phenomena like those described by thermodynamics.

So first, we need to define what is meant by micro and macro. Depending on the scenario, the cutoff between micro and macro can be down at ~10-14 m for lattice vibrations or ~10+1 m for the typical Debye length in the solar wind near Earth.

Next, we must realize that most atomic particles are constantly undergoing some form of oscillation due to thermal fluctuations, zero point vibration, or orbital motion. The typical spatial and temporal variations in atomic processes occur on spatial scales of 10-10 m or less and temporal scales between 10-13 s and 10-17 s. The question is whether we need to worry about these variations or can we find some way to "absorb" or "average" them away. Most instrumentation will unintentionally "average" out most of these variations resulting in relatively smooth and slowly varying (compared to the scales above) macroscopic quantities. This averaging is largely unavoidable for any device seeking to measure macroscopic quantities but it begs the question of what types of averages we should be performing.

The answer turns out to be that we should be performing spatial ensemble averages, e.g., a time average of a simple harmonic oscillator (SHO) will not asymptote to a constant value. That is, the temporal ensemble average of a SHO does not exist but spatial ensemble average average does. Note also that a simple time average always exists. Whether the time average has a useful meaning is another question (see discussion of ergodicity at https://physics.stackexchange.com/a/340874/59023).

We define the spatial ensemble average of some function, $F\left( \mathbf{x}, t \right)$, with respect to some test function, $f\left( \mathbf{x} \right)$, as: $$ \langle F\left( \mathbf{x}, t \right) \rangle = \int \ d^{3}x' \ f\left( \mathbf{x}' \right) \ F\left( \mathbf{x} - \mathbf{x}', t \right) \tag{0} $$ Note that we need not explicitly define $f\left( \mathbf{x} \right)$ a priori, but we do require it be continuous such that it has a rapidly converging Taylor series for distances comparable to atomic scales. Without loss of generality, we can also show that: $$ \begin{align} \partial_{j} \langle F\left( \mathbf{x}, t \right) \rangle & = \langle \partial_{j} F\left( \mathbf{x}, t \right) \rangle \tag{1a} \\ \partial_{t} \langle F\left( \mathbf{x}, t \right) \rangle & = \langle \partial_{t} F\left( \mathbf{x}, t \right) \rangle \tag{1b} \end{align} $$ where $\partial_{j} = \tfrac{ \partial }{ \partial x_{j} }$ and $\partial_{t} = \tfrac{ \partial }{ \partial t }$. This results because the partial differential operators commute through the integral and they are of the ordinate $\mathbf{x}$ not $\mathbf{x}'$. Note you can also use a spatial Fourier transform to derive the macroscopic version of Maxwell's equations [e.g., see pages 248--258 in Jackson [1999]].

Note that spatial ensemble averages are what lets one derive macroscopic properties like the electric polarization, magnetic moment, and the magnetization of a medium. Also note that spatial ensemble averages are not always synonymous with taking the mean of a bunch of spatially separated observations.

Also what the author means by saying that macroscopic variable will become constant in time and spatial averaging?

The spatial averaging is meant to imply what I discuss above, namely, spatial ensemble averages. The time constancy is related to stationary phenomena. That is, we don't want our model to be affected by the individual vibrations or rotations of a single atomic/particle. Note also that the author is throwing out a caveat stated as a limit. That is, they are stating that processes occurring on temporal scales of ~10-7 s or less are considered slow compared to atomic processes. If we start with ~10-7 s being defined as slow, then ~10-5 s is effectively stationary relative to atomic processes.

Why does this matter? Well if we only observe/measure "slow" or "stationary" processes, then we need not concern ourselves with the nitty-gritty details of those super fast atomic processes mentioned above. We don't want atomic processes to affect our macroscopic variable definitions (e.g., we don't want the vibrations of atom A to alter our definition of, say, temperature).

References

  • J.D. Jackson, Classical Electrodynamics, Third Edition, John Wiley & Sons, Inc., New York, NY, 1999.
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