Nonequilibrium Statistical Physics - Science topic

Yes, Prof. , you are right. Perturbation theory works sometimes, but we must agree that there is not a unique approach to solving the kinetic equation.

I apologize for the late reply.

The reason why perturbation does not work always is that there are many physical phenomena that involve processes out of equilibrium with chaoticity and randomness, in addition to.

So sometimes if the problem can be linearized, then perturbation can be used, but sometimes.

Let us see, for example, superconductivity by only considering the T different from the 0 K case (Matsubara frequencies is usually and correctly named) and only taking into account the first term in the zero temperature energy gap (that takes into account self-consistency, or another example is the Vlasov equation in the case of linearization for plasmas is a wonderful example of linearization where perturbation theory applies as you stated.

Best Regards.

Dear Prof. Bruno Cessac, in addition to all the interesting answers in this thread, there is a paper that explains historically the evolution of the Ehrenfest classification of the phase transitions, it might be good to add it since it talks about the Pippard extension of the classification when there are singular points in the specific heat at the T_c as in the ferromagnetic/antiferromagnetic-to-paramagnetic transitions in Ni (ferrom.), MnO (antiferrom.) and other crystals.

Dear Franklin Uriel Parás Hernández Some comments about your interesting thread:

1. At very low temperatures, entropy behaves according to Nernst's theorem

I copy the wiki-web inf. but you also find the same information in Academicians: L. Landau and E. Lifshitz Vol. 5 Vol:

The third law of thermodynamics or Nerst theorem, states that the entropy of a system at zero absolute temperature is a well-defined constant. Other systems have more than one state with the same, lowest energy, and have a non-vanishing "zero-point entropy".

2. Lets try to put Delta Q = m C Delta T, into the expression: Delta(S) >= Delta(Q)/(T2-T1) . What we do obtain? something missing then?

you see, physical chemistry and statistical physics look at entropy in a different subtle way.

3. Delta S = Kb Ln W2/W1 where W is the total number of micro-states of the system, then what is W1 and W2 concerning Delta S?

4. Finally, look at the following paper by Prof. Leo Kadanoff concerning the meaning of entropy in physical kinetics (out of equilibrium systems): https://jfi.uchicago.edu/~leop/SciencePapers/Entropy_is3.pdf

When you refer to thermal gradients you propably refer to solidification processes. For these the temperature gradients are very important as the interface velocity is usually determined crucialy by the transport of the latent heat. Here, of course, special literature exist, but beyond what is taught in general lectures I cannot help much. Your reference to "metallurgy books" seems that there you looked at the soldification sections.

If you are more interested in solid-solid transformations, usually the thermal gradient is less important, simply becausee the latent heat associated with the transformations is much lower than for solidification (the invariant parameter is actually the transformation entropy, connected with the transformation enthalpy via the equilibrium transformation temperature). In that case there is a huge amount of concepts to describe phase transformation kinetics. Porter Easterling is a good starter (having also solidification sections). Models are of course on very different levels. If you want to find much more, you may refer to Christian.

Yes, Patrick Linker .

I believe we do observe quantum tunneling in material with high collision frequency but they will not be observable or not at least observable with our current technology. During the collision the time gap will be extremely small and thus will affect the emission of collided substituents. As their collision will create disturbance in surrounding and thus this disturbance will create favorable condition for creation of matter antimatter pair, which will diminished as breaking virtual particles.

I just read this interesting and lively discussion on negative temperatures. I must confess that I have not read the paper by Abraham and Penrose and I promise to do so during the holidays. I am glad that discussions on the pure thermodynamic issue has arisen, independently of the entropy formulae of statistical physics (by the way, in my view, there is only one “entropy”, Boltzmann’s, as discussed say in Landau and Lifshitz, but that’s another discussion!). So, it is nice to hear arguments based on thermodynamics only.

Although I have not been able to follow the whole thread, I think I side with Struchtrup point of view. I have always had trouble with extending “equilibrium” concepts (such as entropy and temperature) to non equilibrium and/or metastable states. I believe it is dangerous. Certainly, there are many situations where there are no ambiguities, but when there are one should refrain to simply extend the well founded equilibrium concepts, with all their assumptions … About ten years ago there were claims in respected journals about negative heat capacities in nano systems, for instance.

I do not think I have much to say here now, except to recall that, indeed, in order to accommodate for negative temperatures, Ramsey had to change Kelvin-Planck statement of the Second law. Actually, he inverted it (this, I pretended to explain in the appendix of my paper). Then, of course, with the assumption of negative temperatures reservoirs, everything logically follows. However, before arguing about their stability, something very strange happens if equilibrium negative temperatures exist: heat can be converted into work as a sole result (with efficiency one), and the opposite is impossible! hence, there is no need for Carnot engines at all and friction can’t happen! We should be suspicious of this … the argument that reservoirs at negative temperatures are unstable allows us to find the root of the failure, namely, the tacit assumption by Ramsey that stable negative temperature reservoirs exist.

Certainly, the states created in the experiments by Purcell etal, and the many later on, do exist, they are metastable states and can have very long relaxation times to “normal” equilibrium states. But this does not suffice to say that they are in equilibrium for that time, and less to say that they have actually achieved negative temperatures. It may be useful to use those terms but it may also be deeply misleading.

Yes

Another criticism of the Edgeworth (and Gram-Charlier) series is that they do not always (usually?) converge--they are similar to asymptotic expansions).

Instead of expanding the density, you could expand a function that transforms it to normality. The Cornish-Fisher expansions (for such a transformation and its inverse) achieve this. These are based on the Edgeworth expansion. They might also fail to converge but still give approximations.

write to

SATISH TURKANE PATIL "<satish_turkane@yahoo.co.in>

First of all, there isn't any ``paradox''. Next the KAM theorem doesn't provide a way for obtaining the recurrence time of any system-it implies that, under certain assumptions, this time isn't infinite, that's all. How to actually compute it is another issue. So the answer is that, while the KAM theorem may apply to the FPU system, what it implies may not be useful for addressing certain issues. The reason is that while the theorem implies the existence of tori-that describe the periodic motion, whose period is the recurrence time-it doesn't provide any way of expressing the tori in terms of the original phase space coordinates, i.e. of constructing them.

First, let me stress that the activation energy is not a well defined microscopic quantity, but a convenient parameter in which we can hide our ignorance on the details of the many different possibilities for the individual reactions. 

This having been said, the importance of quantum effects at a given temperature can be assessed by comparing the corresponding thermal energy with the characteristic energies for the different statistics, that is, the zero point energy for enclosing the particle in a given volume, which, for Fermions is known as the Fermi energy and for Bosons as the condensation energy.

As an example, let us take the Fermion case. For electrons in sodium metal, with a particle density of 2.65 x 10^28 /m^3, the Fermi energy is 3.24 eV and the Fermi temperature is 37'700 K, proportional to (N/V)^(2/3) and inversely proportional to the mass of the particle. I am not an expert in solution chemistry (if the concept makes sense  at all at such low temperatures), but I think it is fair to assume that, just on steric grounds, the number density of the reacting molecules is at least two orders of magnitude smaller, which divides the sodium result by a factor of the order of 20. Then, we know that the mass of the nucleon is 1836 times larger than that of the electron, and already for a small molecule like methane, this means division by further factor, this time of the order of 30'000. So you see that the degeneracy temperature at which quantum effects become important is below 0.1 K. Exactly the same reasoning holds for the boson case.

In summary, there is no need to change anything in your treatment.

Best regards, René Monnier

One possibility which is worth considering (but no guarantee that it applies to your case), is that by changing the domain size you alter the range of unstable wavenumbers, so that some may disappear for certain ranges of box size. This is a common phenomenon in Vlasov-like equations. You could try to perform such an analysis for your particular case, and see if it explains the behavior.

While not as specific as the answer above, I would look at books on non-equilibrium thermodynamics, particularly those discussing the Onsager reciprocity relations. Under a thermal gradient, the diffusion coefficient contains both an ordinary Fick's law type component but also a, usually an order or more magnitude smaller, thermodiffusion coefficient which is an additional diffusion gradient induced by nonequilibrium coupling between the thermal gradient and the concentration gradient. This is the coefficient in the Onsager reciprocity relation which is the cross-term in the system of two equations (i.e. the right diagonal X12 and X21 components of the thermal force coefficient matrix; X11 and X22 are the normal Fickian diffusion and Fourier heat equation coefficients)

I believe that "conditional Lyapunov exponents" and "transversal Lyapunov exponents" are the same quantity. If so, you want to compute all Lyapunov exponents associated to transversal directions to the synchronization manifold (S). Since you are dealing with a master-slave coupling, the linearised system (which provides the master stability function, obtained by linearising the vector field around a typical trajectory embedded in S) is block triangular. Then the conditional Lyapunov exponents would be given by the eigenvalues associated to the lower-right block (\partial y(t)\partial y). If your coupling function is simple enough, then you probably can find an algebraic expression for the conditional Lyapunov exponents in terms of the (ordinary) Lyapunov spectra of the x dynamical system. You can numerically compute these using the above mentioned Wolf's algorithm. If you wish, you also can compute the Lyapunov spectrum of the full system (dim = dim(x) + dim(y)) and exclude from your analysis those Lyapunov exponents of the driver. Basically, if the driver dynamic has n positive Lyapunov exponents (n=1 for a chaotic system and n>1 for a hyper-chaotic system), it should be sufficient compute, by Wolf's algorithm or any other, the n+1 largest Lyapunov exponents for a reference trajectory embedded in S. If the (n+1)-th exponent is negative, then you know that all transverse (conditional) Lyapunov exponents are negative. It worth to remember that you should consider "typical" perturbations to the reference trajectory. In other words, let the initial conditions (lets say, z(0)=(x(0),0) ) of the reference trajectory be embedded in S and the initial conditions of all other trajectories be around z(0) but with non-null components on y directions. Technically, answering your question: no, I don't know any specific algorithm to compute conditional Lyapunov exponents, but I believe, as Shekatkar said, it should not be that hard to adapt any existent algorithm that computes ordinary Lyapunov exponents.

General analytical solution of 3-body problem can not be obtained in principle as the general solution is unstable (highly sensitive to small changes of the initial conditions). The understanding of this fact lead to the Dynamical Chaos paradigm and development of statistical methods of analysis of such solution (e.g. S. M. Ulam, On some statistical properties of dynamical system, 1961 and 55 years of active investigations)

"life was really a process", "life on Earth was an open system."

First impression is that somebody messed up definitions or did not care about them. A system can be a lot of things. Here, one could think of system in the sense of thermodynamics or more philosophical of the concept from system theory. Either way, a system is surely not a process. Also, things like "almost a generation ahead of his peers" are usually unjustified praise soly intended to impress the reader. Call me close-minded, but after this short paragraph I don't think anything interesting will follow. Probably the usual vioaltion of the concept of entropy.

Actually, a time-homogenous Markov Chain rigorously converges to a unique probability distribution, which can be fixed e.g. by the detailed balance condition. If this distribution is chosen to be the Boltzmann distribution, the MC converges to the Gibbs equilibrium measure, i.e. thermal equilibrium. You can also choose this distribution to be something completely different, e.g. a steady-state non-equilibrium distribution corresponding to some physical system. This is how you make a connection to non-equilibrium thermodynamics.