iMechanica - thermodynamics of nanoscale small systems - Comments

Re: Nanothermo: Thermodynamics of small isolated systems

Biswajit Banerjee — Sun, 20 Nov 2011 21:35:37 -0500

An interesting paper on this subject was published by Mandelbrot in 1962. Check out

http://projecteuclid.org/euclid.aoms/1177704470

The Role of Sufficiency and of Estimation in Thermodynamics

Benoit Mandelbrot

Source: Ann. Math. Statist. Volume 33, Number 3
(1962), 1021-1038.

Abstract

The purpose of this paper is to point out (and to
use) the relations of certain statistical concepts with "statistical" thermodynamics.

(A) It is observed that Gibbs's "canonical distribution"of energy is precisely what statisticians have later labeled a "distribution of the exponential type". It follows that a rigorous treatment of the canonical law can be based upon the concept of "sufficiency", which is thereby related to the physical idea of "thermal equilibrium" and to the "zero-th principle of thermodynamics". In other words, the theory of physical fluctuations can be based upon "principles" very similar to those of the "phenomenological", or "classical, non-statistical" thermodynamics. Naturally, our results will be less detailed than those of statistical mechanics. However, the foundations of the latter theory still raise a host of unanswered
problems, and it seems good in the meantime to show that the less powerful phenomenological theory has a wider scope than is commonly thought (see also [15]). The possibility of a purely phenomenological approach to statistical thermodynamics is not in itself a new idea. A procedure somewhat similar to ours has indeed been long ago suggested in Szilard's admirable, but very difficult and neglected, paper [18]--not to be confused with his [19]. Of course, Szilard used a quite different vocabulary; but, with hindsight, one may now say that he has co-invented the concept of sufficiency with R. A. Fisher; by showing that, under certain regularity conditions, Gibbs's canonical law is the only probability distribution with a single scalar sufficient statistic, Szilard also anticipated the results of G. Darmois [2], B. O. Koopman [10] and E. J. G. Pitman [16], but was partly anticipated by Poincare [17].

(B) The second thesis of the paper is independent of Szilard, and concerns the concept of temperature. For systems with a canonical
energy, the temperature is the parameter of the Gibbs distribution; as such it is undefined for isolated systems with a determined energy. However, it is necessary to generalize the concept of temperature to isolated systems. Several definitions have been proposed and, although they all safely converge mutually for the usual very large systems, the temperature remains athematically ambiguous for small isolated systems; it also becomes physically meaningless. We shall show that the temperature for systems-in-isolation should be viewed as a statistical estimate of the parameter of a conjectural canonical distribution, from which the presently isolated system may be presumed to have once been drawn. This interpretation explains the nature of the ambiguity of the concept of temperature; it also meets the actual practice of physicists; finally, some of the a priori conditions, which the physicists impose upon their "estimators", turn out to correspond to the statistical conditions of consistency, unbiasedness, and efficiency. Physicists also use two very interesting variants of consistency and unbiasedness, which we shall study under the names of "self-consistency" and "self-unbiasedness". The most commonly used temperature, due to Ludwig Boltzmann, turns out to be the maximum likelihood estimator. In summary, we hope to show that it is a great pity that mathematical and physical statistics should have developed largely independently of each other, while using the same concepts. By combining the rigor of modern statistics with the intuitive vigor of thermodynamics, both should be served well. However, as things stand, the mathematical statistician should not hope to unearth in the literature of physics any result as yet unknown to him. An important open problem suggested by this paper is the following. When sufficiency and estimation are defined in the most general terms, it seems that one should also be able to generalize the scope of thermodynamics. However, an approach such as that of P. R. Halmos and L. J. Savage [5] could not be applied to thermodynamics without substantial restrictions, as we shall show in Section 7. It remains to study these restrictions in greater detail, before one can assert that a non-void generalization of thermodynamics is possible. The problem is addressed to both mathematicians and physicists. We shall strive to reduce to the minimum the detailed knowledge of physics required to read this paper. If the reader's appetite for information about thermodynamics has been awakened, he could do no better than to make use of references [11] and [20].

Nanothermodynamics

Libb Thims — Sun, 26 Dec 2010 11:25:41 -0500

I skimmed over much of this discussion, but I would suggest you read up on the work of the pioneers in this field, e.g. Terrell Hill, Ali Mansoori, also possibly Gian Beretta’s thermodynamics
textbook, and his work on single particle system quantum thermodynamics variables,
if I remember correctly. Here’s a starter page:

http://www.eoht.info/page/Nanothermodynamics

Also authors who publish on so-called violations of the second law are all individuals who have never read Clausius' textbook and are aiming to disprove the Boltzmann version of the second law:

http://www.eoht.info/page/Violations+of+the+second+law

temperature without thermal contact

Amit Acharya — Thu, 11 Nov 2010 10:12:40 -0500

Henry:

For an isolated ergodic Hamiltonian system with kinetic energy quadratic in the momenta (all standard assumptions), there is a clear definition of the temperature based on the derived result of equipartition. This does not require the notion of thermal contact for the definition, and an entropy for such a system can be defined (non-extensive) to be consistent with the macroscopic thermodynamic relation between reciprocal temperature and the derivative of the entropy w.r.t energy.

Now how well this definition corresponds with the 'level of hotness' we physically perceive, and think we measure and also think as understand as temperature, that I have not decided in my mind with certainty (I think the question may be undecidable), but there is a clear definition to hang on to without involving thermal contact.

From having skimmed over this voluminous thread, it seems that many here do not like this definition (most must be aware of it since this is a standard notion?).

I would like to know any thoughts on this anyone might have.

Incidentally, in the attachment to the post

http://www.imechanica.org/node/9289

I summarize some basic statistical mechanics in the microcanonical ensemble, all learnt from the book by Berdichevsky referred to in the paper.

I recommend the book highly to students of mechanics - Berdichevsky, being a student of Sedov's and having degrees in both solid and fluid mechanics writes like a mechanician, and after a long search I finally found a book where the fundamentals are laid out clearly - no philosophy. He also makes the connection between statistical mechanics in the microcanonical and canonical distributions in the limit of large N and there are nice connections with when fluctuations become important, a fact that can be captured by working in the microcanonical setting.

- Amit

Thermodynamics of irreversible processes

Zhigang Suo — Tue, 01 Dec 2009 07:28:00 -0500

These are excellent questions, but generally useful answers do not exist. They belong to nonequilibrium phenomena.

For one class of nonequilibrium phenomena, the system is not in equilibrium as a whole, but the system can be divided into small elements, each element being in a state of equilibrium. One can follow the thermodynamics of irreversible processes to anaylze how the system as a whole approaches a state of equilibrium.

A classical example is conduction of heat in a solid. While the entire body is not in a state of equilibrium, each small element of the body is in its own state of equilibrium. It is not meaningful to talk about the temperature of the entire body, but one can talk about the temperature of each element of the body, and apply the equilibrium thermodynamics to every element. One then describes the conduction of heat by using a kinetic law (i.e., Fourier's law).

As another example, in my course on advanced elasticity, I describe nonequilibrium phenomena in gels.

time

Henry Tan — Tue, 01 Dec 2009 04:26:47 -0500

When saying that a system needs “a long time” to be in any of its quantum state, there are some questions:

1) How long?

2) If “time” is involved, it seems that a system needs time to evolve from one state to another state i.e., from one quantum state to another requires a mechanism?

3) During the time that is not that long, what is temperature?

quantum states and the fundamental postulate

Zhigang Suo — Mon, 30 Nov 2009 21:53:45 -0500

The fundamental postulate is also intriguing. It says that a system isolated for a long time is equally probable to be in any of its quantum states. That our world is described by quantum states may be regarded as a law itself.

I read your lecture notes

Henry Tan — Mon, 30 Nov 2009 16:52:00 -0500

Dear Zhigang,

I read your lecture notes on “Statistical Mechanics” posted in http://imechanica.org/node/288. They are very interesting and useful to me. I can understand the effort in reconciling the empirical notions and their statistical meaning.

However I feel that your notes are not complete, or self-sufficient in other word, in giving statistical meaning. The “quantum state” is the fundamental concept in your notes on isolated system, temperature, probability, entropy, free energy, and etc. Yet the “quantum state” itself is undefined.

Regards,
Henry.

Re: statistical meaning of temperature

Zhigang Suo — Mon, 30 Nov 2009 08:46:09 -0500

In teaching a course last spring, I updated my notes on temperature. The central aim of the notes is to reconcile the empircal notion of temperature and its statistical meaning. Hope that the notes help.

statistical meaning

Henry Tan — Mon, 30 Nov 2009 06:05:15 -0500

But the statistical meaning of the temperature is missing, which is the theme of this series of talk.

Welcome to Manchester foundation year students

Henry Tan — Mon, 23 Feb 2009 07:46:47 -0500

Some of the Manchester foundation year students will join in this discussion. Welcome!

Foundation year project title: Thermodynamics of nanoscale small systems

This project explores the thermodynamics of nanoscale small systems and its possible applications. For a system going to the scale of a few (say less than 100,000) atoms, which is possible for today’s technology, one cannot even define the temperature. Is this true or not? This is a bizarre discovery: some concepts for large systems, like temperature, are meaningless in some tiny objects.

"thermal contact" of a one-atom system?

Henry Tan — Tue, 16 Dec 2008 09:48:57 -0500

For a one-atom, or several atoms, system, how are you going to define the “thermal contact”, which is the kernel of the concept of temperature?

How can we use the concept of temperature for few atoms

Teng zhang — Tue, 21 Oct 2008 05:35:48 -0400

Very interesting topic!

I read the first parts of the book Thermal physics written by C.Kittel under the suggestion of Prof. Suo. I post this comment to explain what I learn form that book.

First, the thermal contact plays a signifcant role in the definition of the temperature and entropy. As for two systems, we can obtain the total degeneracy of all the accessible configurations after the thermal contact. If the number of particles in at least one of the two systems is very large, the numbers of that total configurations can be replaced by the number of the states in the most probable configuration. Only in this case, the additivity of the entropy is valid.

As defined in the lectures of Prof. Suo:1/T = change in the logarithm of the number of quantum states divided by the change in the energy of the system, everything else being fixed. The introduction of temperature is to describe the equilibrium state of two systems under thermal contact. It is noted that this equilibrium state is just the most probable configuration. The formalism of T is also derived from the maximum of total degeneracy of all the accessible configurations. In this sense, we can think T is corresponding to the most probable configuration. However, the states expect the most probable configuration can be observed only when the number of particles in at least one of the two systems is very large. If the two systems are both small, then we can see that many different states expect of the most probable configuration which can be represented by the temperature.

Thus, for a small system with only few atoms, we can define the temperature of this kind system via letting it contact with a very large system. However, when we make two small systems together, how can we obtain the final temperature of these two systems, even though we know the temperature of them before contact. If we make they contact with a large system, this may destroy the states of the real systems and make them have the same temperature of the large system itself.

I am not sure whether my understanding is reasonable or not, however, I hope this can make any help for this topic.

Best regards

Teng Zhang

time-temperature equivalence for a viscouselastic material

Henry Tan — Sat, 08 Sep 2007 15:59:24 -0400

How does the time-temperature equivalence for a viscouselastic material be viewed at molecular scale?

How temperature affects the viscoelastic behaviour?

Henry Tan — Sat, 08 Sep 2007 15:56:04 -0400

How temperature affects the viscoelastic behaviour? especially viewed from the behaviour of molecular movements?

Mr Tan, But it

Aneet — Tue, 15 May 2007 10:38:47 -0400

Mr Tan,

But it should make sense, as the molecules/atoms vibrate or move around, they do so because they have velocity. And when we talk about molecular dynamics wont we say KE = (1/2)mv2 where KE = (3/2)NkT (for a 3D system, K - Boltzmann's constant?)

Then we do have energy and thus 'temperature'.

Aneet

thermodynamics of nanoscale small systems

Henry Tan — Thu, 15 Mar 2007 18:29:40 -0400

How to measure the temperature of a nanotube?