iMechanica - statistical mechanics - Comments

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

Thank you zhangzhuo

hitzhangzhuo — Mon, 30 Jun 2008 07:59:13 -0400

Thank you

zhangzhuo hitzhangzhuo@gamil.com

Thanks for sharing. Yi

Yi Han — Thu, 12 Jun 2008 13:00:10 -0400

Thanks for sharing.

Yi Han hyxjtu@gmail.com

How random are "natural" events like IP address?

Ajit R. Jadhav — Sun, 25 Nov 2007 14:42:15 -0500

How random are "natural" events like IP address allocation when the dial-up Internet service provider is an entity like Tata Indicom?

I plan to list here the recent few IP addresses: 126, 45, 22...

I plan to expand this entry later on...

Ajit

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?

Randomness in Molecular Dynamics Simulations

Keng-Wit Lim — Thu, 16 Aug 2007 19:18:49 -0400

Hi Amit,

I am studying nonlinear dynamics and MD. You mentioned about deterministic MD, that the fine dofs evolve by completely deterministic laws.

Consider a very simple MD program. Is it true that there's randomness in MD simulations that is due to, say, the bootstrapping of the simulation using random number generators (initialization of the initial velocities of the molecules)? What I mean is that perhaps the evolution of the dynamics is sensitive to the initial conditions such that the subsequent molecule trajectories may not be predictable? If your initial conditions are always the same, wouldn't you just compute the same answer every time you run the simulation? Perhaps my view is rather simplistic.

You mentioned that the randomness in trajectories of deterministic MD systems is the fact that almost all trajectories visit all neighborhoods of the energy surface. Is the randomness of the MD simulation due to bootstrapping related to this reason?

Thank you in advance.

Keng-Wit Lim

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

A side is a side of something...

Ajit R. Jadhav — Thu, 03 May 2007 10:52:24 -0400

A "side" is not a floating abstraction--it's a side of something. That something, here, is randomness.

Inasmuch as your comments pertained to randomness, they were, and still are, welcome.

Once the discussion started veering towards the details of your particular research but away from randomness, even that was OK, up to an extent. (One cannot expect people to think disconnectedly.) But when you started asking me to identify whether I understood this or that part of *that* research of yours or not, whether that methodology was understandable or not... things of that nature... I thought it was high time to keep the matters straight and to tell what this thread was all about--even if it was a Carnegie Mellon faculty member in question.

Am I happy about this turn of events? Of course not.

Further, I must add that I feel sure you wouldn't quite behave this way if both you and I were not Indians. In my experience, typically, Indians reserve their most polite interactions for others--especially (but not at all exclusively), for white Americans.

But yes, when it becomes necessary, one has to take a firm stand. I did. But it's not a matter of any further importance the way I see it. So, even if I am not happy about it, this matter is of the sort I guess I could forget within a day or two.

I close this discussion with the observation that even while publicly rueing about staying away, you *still* did *not* address the pertinent (and even "mathematical" (LOL!)) issue of dynamical systems mentioned in my point 2.1. Is this kind of communication typical of all Carnegie Mellon faculty members? (I have been wondering if my PhD research would stand a chance to get any funding in the USA. CMU being a "private" sector school, one thinks a little more.)

As to whether the point 2.1 was relevant to this thread in the first place or not, I prefer to submit this entire matter to the scrutiny of a future engineer who might read it with a fresh mind--an engineer who comes years, perhaps decades and centuries later--a future Ajit R. Jadhav, if someone must put it that way out of whatever sort of motivation he has.

Though this correspondence could have been better, I mean it genuinely when I say thanks for your participation anyways. And, yes, goodbye!

If it means saying goodbye to the whole of US researchers community, or the whole world science and engineering community, I couldn't care less.... As I said, in the absence of any proper feedback, I must regard my openly expressed thoughts as addressed to a future engineer with a fresh mind, perhaps decades and centuries later--a John Galt, so to speak (if you have read Ayn Rand's fiction)....

In the meanwhile, I brace myself for idiotic comments to percolate through Yahoo, msnbc.com, Times of India and other media in the days and weeks to come--though, given the sum totality of my knowledge, I can't expect *that* to be there decades centuries later. Metaphysically, it can't last that long, though probably I will have to face it as long as I live. (The last two paras are mostly for others, Dr. Amit Acharya!)

AJ: The core point of this

Amit Acharya — Sun, 29 Apr 2007 22:05:23 -0400

AJ: The core point of this thread is the term "randomness"--its meaning and so on...

In your first post for the forum, I believe you had said you were interested in side comments, brain storming, so I thought providing *a* point of view that relates to some manifestations of randomness wouldn't be a bad idea.

But thanks for your clarification - Iwill know to stay away.

On the technology of producing random numbers...

Ajit R. Jadhav — Sun, 29 Apr 2007 13:54:11 -0400

Often times, chaos theorists (and many others too) claim that randomness is universal. Keeping aside for a moment the philosophical questions such as how (and whether) randomness is (im)possible, what is the meaning of the term, etc., here are a few practical questions pertaining to technology:

Why can't we have a chip that has circuits especially built to produce "randomness"? Something like an electronic version of the lottery apparatus they show on TV. The lottery apparatus relies on the fluid dynamical instabilities, but this one should rely on the classical circuit instabilities. Come to think of it, if an off-tune TV (or radio) shows (or emits) noise so easily, what prevents us from designing and building a "random chip"? If mass-produced, it could get as inexpensive as under a dollar or so!

There are many efficient algorithms that could use a fast random generator. Why not produce it in hardware--and directly?

A question: What would be the estimated frequency at which one could read off the next random number from the registers of such a chip? Does anyone know any principle that sets the upper limit? In other words, shouldn't a randomn number generator of this kind have an "incubation" period before the next random number can be reliably produced by it, and if yes, how short could such a period get?

If this post doesn't generate any thoughts--on randomness, I would better stop blogging here at least for the time being. Seems like this forum is better suited to those people who should have graduated in analytical mathematics but turned out to be "engineers"--oops, "mechanicians"...

It's been about randomness--really!

Ajit R. Jadhav — Sun, 29 Apr 2007 13:20:51 -0400

Hello Amit,

AA: I don't know if your comment on 'generality' refers to what you understood or about the methodology I propose.

--> AJ: In case you really couldn't get it by the context: My comment refers to what I understood by browsing through your post and papers.

Now, if nonlinear dynamical systems theory is an area of your knowlede or expertise, I would like to know if you would want to address questions like the question (2.a) in my further post dated 24th April (see below).

----

AA: The whole point of my paper is not about randomness in coarse behavior.

--> AJ: The core point of this thread is the term "randomness"--its meaning and so on...

apparent randomness continues

Amit Acharya — Tue, 24 Apr 2007 16:41:14 -0400

Hi Ajit,

Response to some of your comments:

AJ - (ii) Characterizing system evolution in terms of averages won't necessarily be deterministic. (You really caught me here!!)

This is if one fixes the set of coarse variables a-priori. If, however, one is prepared to augment this set (according to well-defined procedures in dynamical systems theory) then such apparent randomness can be expected to be removed.

AJ - From your writing, I gather that MD procedures involve dynamic instabilities (or deterministic chaos). Is that so?

Well the nature of randomness in trajectories of deterministic MD systems is the fact that almost all trajectories visit all neighborhoods of the energy surface. Some neighborhoods are visited with higher probability density, but nevertheless all possible states consistent with constant energy are visited arbitrarily closely.

Generally, it is perhaps OK to say that a nonlinear Hamiltonian systems with many degrees of freedom exhibits unstable behavior - there are, of course, many particular examples that are well understood. But we should have some experts weigh in on this.

AJ - About definition of randomness--yours and mine: Frankly, I am not sure if I have *one* definition of randomness. For that matter, I am not even sure if I have *any* definition of this term. In fact, that's precisely the reason why I began this thread! If you have a definition (or any concise description) for this term, what would it be?

My simpleminded operational definition would be this: Generally we expect evolution to be state-dependent. This requires a definition of state. Let's say one has chosen a set of observables whose values at any given time collectively form the state at that time. If for the same state, one sees two or more possible evolution of the state, then I would call this random behavior, in the context I am talking about here.

AJ - After reading your post and paper, what I gather is just a vague point that if you are going to drop *some* information in going from the finer to the coarser scale, then, even if you begin with a deterministic system, the end result is going to carry wild fluctuations characteristic of stochasticity, and there are some interesting mathematical properties about how this happens. Now, this is too general an "understanding" to be useful in science or engineering!!

I don't know if your comment on 'generality' refers to what you understood or about the methodology I propose.

The whole point of my paper is not about randomness in coarse behavior. It is about realizing practically (e.g. algoritmically) how one can systematically make a good choice of a coarse set of variables in terms of which apparent randomness can be avoided and a determinstic theory for coarse response can be posed.

On the latter part (i.e. once a choice of coarse variables has been posed, how to develop the coarse model), my students and I have solved small but hard nonlinear problems - and, by the way, these are problems from science and engineering! If interested see the attached papers.

But the choice of an optimal set of coarse variables is not resolved in these papers and the latest one is an effort to deal with that issue. With my students we have also tried these ideas out on model systems (e.g. Lorenz) with success, so it seems like the ideas will be useful for science and engineering.

AJ: I also think that for the purposes of this thread, perhaps a very simple mathematical model (say, something like the simplest logistic map) might be more than sufficient.

If you are interested, look att he discussion surrounding Fig. 1 in the '...Some examples' paper. If one settled on the x, z variables as the coarse set, then in this state space coarse response seems random at the point of self intersection of the projected Lorenz trajectory, which of course is a completely deterministic system.

Now that I am finished, I realize I do not know how to attach files to a reply. So I just put in a new blog entry.

Some more questions on randomness...

Ajit R. Jadhav — Tue, 24 Apr 2007 12:17:49 -0400

Here are some more things that occurred to me over the past few days.

(1) Consider the fact that there are no random numbers--there are only random sequences. If so, is "randomness" a property of a collection? Sub-questions follow:

(1.a) Any finite sequence of numbers can be plotted as a graph of points in the X-Y plane. For any such a set of points, it's always possible to find an algebraic polynomial which passes through everyone of them. In other words, there is that equation which *predicts* that sequence of numbers. So, no finite sequence can be random. Is this conclusion true or false? How about the argument itself--is it logically sound?

(1.b) Hence, does true randomness require a sequence of infinitely many points? (Here, note, a finite line segment carries infinitely many points too.)

(2) Have a look at this great introduction to the topic of "chaos" by David Harrison: Chaos. Also see a related (low bandwidth) animation hosted on the same site: Animation of 3 body problem These Web pages are accessible to undergraduates. (That is the major reason why the site appealed to me!) The following questions are specifically based on the contents of the above two Web pages:

(2.a) The animation shows how the green and red trajectories soon enough begin to diverge greatly even if their horizontal speed initially differs only by 1 %. Such a divergence is often put forth as a possible model for how randomness might come about via a deterministic evolution.

But it is just as well possible to reverse trace the trajectories. In this case, you start from the two divergent points and proceed to come converge!

The question is: If, in this model, I pick the two points arbitrarily, will their trajectories always come very close to each other sooner or later? If not, why not?

(2.b) Is the "non-repeatability" mentioned in the first Web page (near the graph of the rabbit population at L equal to 0.0000395) the same as "randomness"?

Thanks in advance for letting me know your thoughts, comments, opinions, etc.!

Ideas in Amit's reply...

Ajit R. Jadhav — Tue, 24 Apr 2007 12:11:57 -0400

Hello Amit,

I appreciate the following points made by you:

(i) The finer -> coarse model is just that: a *model*, i.e. just a possible way to show how apparent stochastic behavior might come about from a deterministic system
(ii) Characterizing system evolution in terms of averages won't necessarily be deterministic. (You really caught me here!!)

About MD: I do not have any background on MD except for what an engineer might gather by browsing general articles such as those on Wikipedia and so on. From your writing, I gather that MD procedures involve dynamic instabilities (or deterministic chaos). Is that so?

About definition of randomness--yours and mine: Frankly, I am not sure if I have *one* definition of randomness. For that matter, I am not even sure if I have *any* definition of this term. In fact, that's precisely the reason why I began this thread! If you have a definition (or any concise description) for this term, what would it be?

I had already flipped through your paper of the title "On the choice of coarse variables for dynamics" when that post of yours first appeared. Today, I once again went through it, though I am afraid I cannot understand much of it. For instance, I have no knowledge of the invariant manifold theory (or any other topic from topology). After reading your post and paper, what I gather is just a vague point that if you are going to drop *some* information in going from the finer to the coarser scale, then, even if you begin with a deterministic system, the end result is going to carry wild fluctuations characteristic of stochasticity, and there are some interesting mathematical properties about how this happens. Now, this is too general an "understanding" to be useful in science or engineering!!

I also think that for the purposes of this thread, perhaps a very simple mathematical model (say, something like the simplest logistic map) might be more than sufficient. After all, the main question I am here concerned about is not one kind of dynamical evolution against another but just the basic notion of randomness: its meaning, including the question of whether the term at all carries any meaning or not, and if yes, in what way that meaning could be delineated.... I am also posting a few more questions in this direction right in this thread; please have a look at them too....

Overall, though, let me again say that I have very much appreciated your interest. Thanks!