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# Nonequilibrium statistical mechanics - Science topic

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Dear RG community members, this pedagogical thread is related to the most difficult subject among the different fields that physics uses to describe nature, i.e. the physical kinetics (PK). Physical Kinetics as a subject is defined as a “method to study physical systems involving a huge number of particles out of equilibrium”.
The key role is given by two physical quantities:
• The distribution function f (r, p, t), where r is a vector position, p is a linear momentum and t is the time for the function f which describes a particle in an ensemble.
• The collision or scattering term W (p, p¨) gives the probability of a particle changing its linear momentum from the value p to the value during the collision.
If the following identity is satisfied for the distribution function df (r, p, t) / d t = 0, then we can directly link PK to the Liouville equation in the case that the distribution function does not depend on time directly. Physics students are tested on that, at the end of an advanced course in classical mechanics, when reading about the Poisson brackets.
However, is important to notice that not all phys. syst. are stationary and not always the identity df /d t = 0 follows, i.e., the distribution function - f is not always time-independent, i.e., f (r, p) is just true for some cases in classical and non-relativistic quantum mechanics, and the time dependence “t” is crucial for the majority of cases in our universe, since is out of equilibrium.
In addition, physical kinetics as a “method to study many-particle systems” involves the knowledge of 4 physics subjects: classical mechanics, electrodynamics, non-relativistic quantum mechanics & statistical mechanics.
The most important fact is that it studies the scattering/collision of particles without linear momentum conservation p, where: the time dependence & the presence of external fields are crucial to study any particular physical phenomena. That means that PK is the natural method to study out of equilibrium processes where the volume of the scattering phase space is not conserved & particles interact/collide with each other.
If the phase scattering space vol is not conserved, then we have the so-called out of equilibrium distribution function which follows the general equation:
df (r, p, t) / d t = W (p,p¨), (1)
where: d/dt = ∂/∂t + . ∂/∂r + p´. ∂/∂p, with units of t -1or ω/(2π).
The father of physical kinetics is Prof. Ludwig Eduard Boltzmann (1844 – 1906) [1]. He was able to establish the H theorem which is the basis for the PK subject and also he wrote the main equation (1), i.e., the Boltzmann equation to describe the out of equilibrium dynamics of an ideal gas. & in d/dt are derivatives, p¨ in W is another momentum position
Another physicist who established the first deep understanding and condensed the subject into a book was Prof. Lev Emmanuilovich Gurevich (1904 - 1990). He was the first to point out that the kinetic effects in solids, i.e., metals and semiconductors are determined by the "phonon wind", i.e., the phonon system is in an unbalanced state [2]
Physical kinetics has 3 main approaches:
• The qualitative approach involves the evaluation of several physical magnitudes taking into account the order of magnitude for each of them.
• The second approach is the theoretical approach which involves complicated theoretical solutions of the kinetic equation using different approximations for the scattering integral such as the t approximation. For graduate courses, I follow [8], an excellent textbook by Prof. Frederick Reif. For undergraduate teaching, I followed the brief introduction at the end of Vol V of Berkeley Phys C.
• The numerical approach since most problems involving PK requires extensive numerical and complicated self-consistent calculations.
The fields where PK is useful are many:
• The physics of normal metals and semiconductors out of equilibrium.
• The hydrodynamics of reacting gases & liquids, quantum liquids, and quantum gases at very low temperatures.
• The physics of superconductors, phase transitions, and plasma physics among others.
There is a quantum analog to the classical Boltzmann equation, we ought to mention three cases: the density matrix equation for random fields, the density matrix equation for quantum particles, and the Wigner distribution function. Main graph 1 is adapted from [4] to the English language, LB picture from [7], and LG picture from [3].
Any contributions to this thread are welcome, thank you all.
References:
2. Fundamentals of physical kinetics by L. Gurevich. State publishing house of technical and theoretical literature, 1940. pp 242
3. Lev Emmanuilovich Gurevich. Memories of friends, colleagues, and students. Selected Works, by Moisey I. Kaganov et. at (1997) pp 318. ISBN:5-86763-117-6. Publishing house Petersburg Institute of Nuclear Physics. RAS
4. Белиничер В.В. Физическая кинетика. Изд-во НГУ.Новосибирск.1996.
5. Lifshitz E., Pitaevskii L. 1981. Physical Kinetics. Vol. 10, (Pergamon Press).
6. Thorne, K. S. & Blandford, R. D., Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, & Statistical Physics (2017) (Princeton University Press).
8. Fundamentals of Statistical and Thermal Physics: F. Reif Mc Graw-Hill, 1965
Thank You, Prof. Akram Louiz for your post and the publication. I will read it.
Best Regards.
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I am asking this question on the supposition that a classical body may be broken down in particles which are so small in size that quantum mechanics is applicable on each of these small particles. Here number of particles tends to uncountable (keeping number/volume as constant).
Now statistical mechanics is applicable if practically infinite no. of particles are present. So if practically infinite number of infinitely small sized particles are there, Quantum Statistical Mechanics may be applied to this collection. (Please correct me if I have a wrong notion).
But this collection of infinitesimally small particles make up the bulky body, which can be studied using classical mechanics.
There is no difference Prof. Manish Khare, we have two windows to watch the physical world, the classical & the quantum approaches, but there is a window, they are the Wigner probabilistic distributions.
Best Regards.
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A  phase transition of order k is mathematically characterized by a loss of regularity of free energy f: f is k-1 differentiable but not k differentiable. There are many examples of first and second order phase transitions in experiments and in models. There are also cases where f is C^{\infty} but not analytic (Griffith singularities).
But are their known example of phase transition of order k, k>2 ?
A third order phase transition would mean that quantities like susceptibility or heat capacity are not differentiable with respect to parameters variations. But I have no idea of what this means physically.
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 Tc as in the ferromagnetic/antiferromagnetic-to-paramagnetic transitions in Ni (ferrom.), MnO (antiferrom.) and other crystals.
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Dear Colleagues :
Does anyone have literature referencing the diffusion process of Carbon (I mean Carbon atoms) into Bismuth Telluride (Bi2Te3) or into some other compound alike ? E.g. PbTe, (Sb,Se)Bi2Te3, Sb2Te3, etc ... ?
I'll really appreciate if someone can help me out
Kind Regards Sirs !
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Dear all:
I hope this question seems interesting to many. I believe I'm not the only one who is confused with many aspects of the so called physical property 'Entropy'.
This time I want to speak about Thermodynamic Entropy, hopefully a few of us can get more understanding trying to think a little more deeply in questions like these.
The Thermodynamic Entropy is defined as: Delta(S) >= Delta(Q)/(T2-T1) . This property is only properly defined for (macroscopic)systems which are in Thermodynamic Equilibrium (i.e. Thermal eq. + Chemical Eq. + Mechanical Eq.).
So my question is:
In terms of numerical values of S (or perhaps better said, values of Delta(S). Since we know that only changes in Entropy can be computable, but not an absolute Entropy of a system, with the exception of one being at the Absolute Zero (0K) point of temperature):
Is easy, and straightforward to compute the changes in Entropy of, lets say; a chair, or a table, our your car, etc. since all these objects can be considered macroscopic systems which are in Thermodynamic Equilibrium. So, just use the Classical definition of Entropy (the formula above) and the Second Law of Thermodynamics, and that's it.
But, what about Macroscopic objects (or systems), which are not in Thermal Equilibrium ? Maybe, we often are tempted to think about the Entropy of these Macroscopic systems (which from a macroscopic point of view they seem to be in Thermodynamic Equilibrium, but in reality, they have still ongoing physical processes which make them not to be in complete thermal equilibrium) as the definition of the classical thermodynamic Entropy.
what I want to say is: What would be the limits of the classical Thermodynamic definition of Entropy, to be used in calculations for systems that seem to be in Thermodynamic Equilibrium but they aren't really? perhaps this question can also be extended to the so called regime of Near Equilibrium Thermodynamics.
Kind Regards all !
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
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Dear colleagues,
I try to perform MD simulation of abstract polymer in good solvent. Particularly I try to calculate solution viscosity by means of non-equilibrium dynamics (shearing). Although shear rate is low – 0.0005 (LJ units), polymer chains longer than 30 bids become highly elongated along flow velocity vector. XX component of Rg tensor becomes 10 times larger than YY and ZZ.
Do you have any ideas, why longer chains demonstrate such behavior?
Is there theoretical relation between viscosity and Rg for such case?
MD details:
LAMMPS, LJ units, repulsion only lj potential, FENE bonds for polymer, no angle potential for polymer, box size 20x20x20, total 8000 particles (density = 1), 15% of particles are attributed to polymer chains (10 to 60 particles in a chain), timestep = 0.01, nose-hoover nvt operating on ‘deform-corrected’ temperature computed by temp/deform.
Dear Didier Lairez,thank you for this clue!
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Dear Research-Gaters,
It might be a very trivial question to you : 'What does the term 'wrong dynamics ' actually mean ?'. I have heard that term often times, when somebody presented his/her, her/his results. As it seems to me, the term 'wrong dynamics' is an argument, which is often applicable to bring up arguments that a simulation result might be not very useful. But what does that argument mean in physical quantities ? It that argument related to measures such as correlation functions, e.g. velocity autocorrelation, H-bond autocorrelation or radial distribution functions ? Can 'wrong dynamics' be visualized in terms of a too fast decay in any of those correlation functions in comparison with other equilibrium simulations, or can it simply be measured by deviations of the potential energies, kinetic energies and/or the root-mean square deviation from the starting structure ? At the same time, thermodynamical quantities such as free-energies might not be affected by the term 'wrong dynamics'. Finally, I would like to ask what the term 'wrong dynamics' means, if I used non-equilibrium simulations which are actually completely non-Markovian, i.e. history-independent and out-of equilibrium (Metadynamics, Hyperdynamics). Thank you for your answers. Emanuel
Start by an obvious remark: the fact that two given Markov process tend to the same (given) equilibrium does not mean that they do so in the same way. In particular, their dynamical behaviour may be quite distinct.
Now Newtonian mechanics will generate some kind of equilibrium. It is often easier to reach that same equilibrium by a stochastic process (Monte-Carlo). Then we are guaranteed that the equilibrium properties will in fact be the same. Static correlation functions, for example, will be correct. However, the dynamical properties need not. Thus the velocity autocorrelation function (the product of v at time t with v at time t+ tau averaged over t) need not have any clear connection. Indeed, in some MC models, there are no velocities!. To the extent that the system is classical, the Newtonian dynamics is the correct one'' and the MC is fake. The results that should be trusted are thus the Newtonian ones.
However, in many cases, MC will give dynamics that are, in some sense, qualitatively close to what Newtonian dynamics gives. Nevertheless, such issues must be treated with considerable care.
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Hi everyone!
Where can I find numerical or experimental estimations of critical exponents for the O(2) symmetric phi^4 field theory in two dimensions ( I mean, for instance, a two dimensional Bose liquid)?
One can find the numerical estimations for the XY -- model; but nevertheless, are there the direct ones specifically for the  phi^4 field theory?
Thx.
The books present indeed the methods for calculating them. The two-dimensional O(2) model is the (two-dimensional) XY model, they're the same model. One should be careful to distinguish knowing by heart'' from understanding''. Cf.  http://www.mit.edu/~levitov/8.334/notes/XYnotes1.pdf for some more background. One point is that the singularities are not, necessarily power laws; the correlation length has an essential singularity at the critical temperature, therefore the exponent ν isn't defined as usual. Therefore one should first learn the theory and find out whether critical exponents and which ones can be defined and, therefore, calculated. The same statements hold for the experiments, too-those that have probed the relevance of the Berezinskii-Kosterlitz-Thouless scenario don't focus on critical exponents, but on other quantities.
The theory was originally worked out by Berezinskii, Kosterlitz and Thouless and experiments on superfluid films have, also, been done. (Some of) the literature is discussed in Zinn-Justin's book in the corresponding chapters. Cf. also Polyakov's book and papers from 1975. Most experiments have been done on superfluid or superconducting films and it's quite straightforward to search for them; it does requires some effort for picking out the most relevant.
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In T-V diagram, process is carried out by increasing pressure. As we increases pressure, specific volume of saturated liquid is larger and specific volume of saturated vapor smaller. What is the reason?
A good question! Not sure I really know the correct answer. In the 2 phase part of the phase diagram there is only one thermodynamic parameter required to define the state of the fluid with wetness. This zone extends upto and including the phase boundary. So as the saturated fluid moves along the boundary not only does its pressure rise but also so does its temperature. This must be the origin of the growing specific volume you draw our attention to. Put simply the temperature increase outweighs the pressure rise. But what i don't really understand is physically why the liquid expands. I guess the story on the saturated vapour side is the inverse the gases must loose heat to remain saturated, entropy falls and the specific volume reduces.
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The idea is to learn how to solve Boltzmann equation for more complicated cases: N electrons considering fields (electric and magnetic) and many scattering mechanism. In this way it is possible for example to calculate mobility, drift velocity of the carriers in devices. There are a lot of books about this topic, however none of them show examples of the code, even for a simple case.
Download freeware code with source. For example, EGS or Penelope. After that read manual and edit the code as you wish.
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The thermal rate coefficient can be obtained from the reactive cross section (σ(Ecoll)):
k(T) = c(T)×∫P(T,Ecoll)Ecollσ(Ecoll)dEcoll
where Ecoll is the relative collision energy and c(T) is a constants at a given temperature and P(T,Ecoll) is the statistical weight.
In normal case Boltzmann statistic is used for the calculation of statistical weights. But Boltzmann statistic is valid when the temperature is high and the particles are distinguishable. At ultralow temperatures (T< 10K) we should use the appropriate quantum statistic (Fermi or Bose).
What kind of quantum statistic should be used in the collision of a
radical[spin = 1/2] + closed shell molecule (spin=0)
at ultralow temperatures?
What is the form of P(T,Ecoll) in this case?
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
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I'm studying an wave system inspired Hamiltonian model with short and long interactions. In particular, I've found a curious size dependent phase transition depending on how strong is my short-range coupling. When N is large, and the short-range coupling remains the same of the small N (where I've found the phase transition), the system remains homogeneous. There is a simple explanation for that?
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.
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For a molecular dynamics simulation , which is the time scale on which we can see the Soret effect a suspension of nanoparticles dispersed in an explicit solvent ?. I just want an estimate , how could make a crude estimate before running the simulation?
A guess may be made by comparing the thermal velocity of explicit liquid by the temperature gradient with the asymmetry of the molecular forces on the nanoparticle at the interface of continuum sustaining velocity gradient.
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Suppose a fluid is in between two walls at different temperatures , each wall is composed of a fixed number of atoms , a fixed volume and a fixed temperature (gas ) , replicating the NVT ensemble for each wall.
Suppose further that the presence of these walls at different temperature induces a linear temperature gradient on the fluid contained. Assuming the fluid is atomic and integrating the equations of motion is performed with the algorithm verlet -velocity .
My question is what statistical ensemble is playing on the fluid confined in these conditions?
If anyone has any reference would be grateful.
The fluid is out of equilibrium and cannot be described by any ensemble. On the other hand, if you have, as you assume, a linear T profile, you should probably have, too a very good approximation, that local thermal equilibrium holds, that is, for every sufficently small  volume element, we have a given position-dependent temperature and chemical potential, and the particles in that volume elemen are in a grand canonical ensemble.
I and some others performed a simulation very similar to what you seem to have in mind in Larralde et al. J. Stat. Phys 113, 197 (2003). Best wishes.
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Let x and y be the state vectors of two dynamical systems, respectively and we have:
dx(t)/dt=F(x(t))
dy(t)/dt=G(y(t),x(t)),
where x is the driver system and is the response system.
According to the literature (e.g. http://journals.aps.org/pre/abstract/10.1103/PhysRevE.61.5142), given a certain coupling strength between the driver and the response, the maximal Lyapunov exponent (correction: it should be conditional Lyapunov exponent) is negative when the response system synchronizes with the driver system. See the attached figure (from the reference paper).
Now my question is: What algorithm should I use to calculate conditional Lyapunov exponent for the purpose of detecting synchronization?