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61

When you say "why aren't things being destroyed", you presumably mean "why aren't the chemical bonds that hold objects together being broken". Now, we can determine the energy it takes to break a bond - that's called the "bond energy". Let's take, for example, a carbon-carbon bond, since it's a common one in our bodies. The bond energy of a carbon-carbon ...

17

$\hbar$ does not need to appear in classical statistical mechanics. You are free to replace it with any quantity with units of angular momentum, say $\hbar_{\mathrm{C}}$. As long as this is choosen smaller than the size you can experimentally probe (i.e., as long as you don't ask questions of the theory that contain structure on this length scale or below) ...

6

I can see different subtleties in Landau's argument. First of all, it isn't entirely clear what is meant by "there are only seven additive constants of motion". To give an example, consider a single particle hamiltonian: $$H=\frac{\mathbf p^2}{2m}+m\omega ^2\frac{\mathbf q ^2}{2}.$$ For this hamiltonian there are several conserved quantities: $$e(\mathbf ... 5 Giving the value simply of k_B T is generally more useful, because I can plug that into anything. Sure, I might need to know the ideal gas energy, and multiply by 3/2. But maybe I need to put it into a partition function, and I just need k_B T. Maybe I'm worried about a harmonic oscillator and I just have the two degrees of freedom. The 3/2 is ... 5 h factor The factor of 1/h^{3N} is a total hack. The integral over phase space has dimensions, whereas Z only makes sense if it's dimensionless. The h factors are there to make Z dimensionless. Suppose you have a system with only one particle in one dimension. Then the integral in phase space goes over one position variable, dq and one momentum ... 4 First let me repeat what Yvan Velenik says in the comment: The terminology is somewhat unfortunate, because you don't need that much statistics, rather you'll need some probability theory. To elaborate, quoting Wikipedia, Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. [...] Statistics deals ... 4 Characteristically, a critical point occurs somewhere anytime you have a continuous phase transistion. That is, if you have two phases of a substance that themselves share their intrinsic symmetries. The classic example is the critical point associated with the liquid gas transition, as you note. Liquids are isotropic and homogenous, gases are isotropic ... 4 In statistical mechanics, at least when you can ignore the spin of the particles you're dealing with, the occupation number of a quantum state (that is, the number of particles in the state, or probability of a particle being found in the state) is proportional to e^{-E/kT}, where E is the energy of the state. If you have a harmonic oscillator at ... 3 The thermal energy k_{B} T is really referring to the probability of finding a system in a state of energy E, given that it is in a surrounding enviroment at temperature T. This probability is proportional to e^{-E/(k_{B} T)}. Using this you can derive a great many things, including the Boltzmann/Fermi distributions. The proportionality constant is ... 3 First of all you can directly write p as a function of V without solving a quadratic equation:$$ p = \frac{N k_B T}{V- N b } - a \frac{N}{V} $$Then you take a look at the interesting points$$\left(\frac{\partial p}{\partial V}\right)_{T,N} = 0 $$The solutions to this equation tell you where the bulk modulus (or its inverse, the compressibility) ... 3 It seems to me that you just cannot tell the difference between a Bose condensate and nothing in this case. What will change if you add some photons or phonons with zero energy to the system? No characteristics of the system will change. So it seems to me we have no criterion to decide if there is a Bose condensate in this case, and what's more important, it ... 2 You will probably find the Stirling approximation useful here. For your purposes it will be sufficient to use the form \ln N! \approx N\ln N - N, which is valid for N\gg 1. 2 When calculating expectation values, you need to know a few things: What is my random variable? What is my distribution function? What is my desired quantity in terms of the random variable? The general form in one dimension would look like this$$ \langle G(x) \rangle = \frac{\int_{x_{min}}^{x_{max}} f(x) G(x) dx}{\int_{x_{min}}^{x_{max}} f(x) dx} \, ...

2

Now, in my book there is not this factor $1/h^{3N} N!$. Where this factor comes from? Why do we need to include it there? $\mathbf{1/h^{3N}}~$: Some people like the value of the partition integral to be independent of units, so they add the denominator $h^{3N}$ to the formula where $h$ has the same units as $pq$. This makes the expression dimensionless ...

2

Things actually do get destroyed by what those air molecules pick up and throw around. Take look at this example [image from here: http://en.wikipedia.org/wiki/File:Arbol_de_Piedra.jpg ] Just like their bigger sized brothers, it's the load of those mini-torpedos that brings the destruction.

2

The partition function is not in general equal to one. It is a normalization constant, i.e. the probability of being in a configuration $\{\sigma\}$ is $$P(\{\sigma\}) = \frac{e^{-H(\{\sigma\})/k_B T}}{Z}$$ where $$Z(T) = \sum_{\{\sigma\}} e^{-H(\{\sigma\})/k_B T}$$ Since multiple configuration can have the same energy, you can define (with a little ...

1

For a classical Maxwell-Boltzmann gas, the partition function is given by $$Z=\sum_i g_i e^{-\beta \epsilon_i}$$ where $g_i$ is the degeneracy. And the probability for each level to be occupied by one particle is given by $$P_i=\frac{g_i e^{-\beta \epsilon_i}}{Z}$$ The partition function function in the above expression is essentially a normalization factor ...

1

Normally we do NOT calculate the phase space density of a system. In the phase space formulation of classical statistical mechanics, the phase space density $\rho(p,q;t)$ has its specified form for different ensembles. Normally for systems at equilibrium the density $\rho$ has no explicit time dependence and thus we work with $\rho(p,q)$. (1) For ...

1

I think what they could mean is that $\vec{\alpha}\vec{\alpha}$ is a second rank tensor that is contracted with $\overline{g}$. I saw this notation being used in the context of electrodynamics before. It is used to get a simple notation for multi-dimensional Taylor series. So we get $$\Delta S=\overline{g}:\vec{\alpha}\vec{\alpha} \equiv \sum_{ij} ... 1 There are multiple ways to justify this identification. The first one is to recognize that in thermodynamics the Helmholtz free energy is the right thermodynamic potential to figure out spontaneous evolution of your thermodynamic system (in virtue of the second principle of thermodynamics) at fixed (N,V,T). In statistical mechanics, this is expressed by ... 1 Normal fluids at fixed pressure are liquid (or solid) at low temperature and gaseous at high temperature. At some temperature the phase transition from liquid to gas occurs. However, at sufficiently high pressure, there is no such phase transition. The critical point is the largest pressure and temperature where the phase transition can be (just barely) ... 1 A proper derivation of the Boltzmann equation from non-equilibrium quantum field theory (which will give the factors 1\pm f in the weak coupling, quasi-particle dominated, limit) is a difficult problem. The standard reference is Kadanoff and Baym, Quantum Statistical Mechanics. The standard approach in introductory text books (and indeed, historically, ... 1 Partition functions are a measure of the allowed volume in (microscopic-)configuration space for the system, and as such they are the normalizing function for probabilities expressed as volumes in configuration space (and assuming the applicability of the ergodic hypothesis). I know that this is very abstract, but it is also very general. 1 I found it:$$\bar{n}_{BE}=-\frac{1}{\mathcal{Z}}\frac{\partial \mathcal{Z}}{\partial x}$$where x =(\epsilon -\mu)/kT$$\mathcal{Z} = \sum\limits_{n}^\infty e^{-n(\epsilon -\mu)/kT}$$which converges if \epsilon -\mu<0 1 You are considering a system of fermions (SC hamiltonian) and a system of bosons (weakly interacting BEC). In order for the density to be finite, fermions must have a positive chemical potential \mu>0. On the other hand, the chemical potential of a system of bosons is less or equal to the energy of the lowest-energy state. In a non-interacting system ... 1 Here's a way to get a decent distance estimate for solids, liquids or gases.: for solids or liquids, you can get the number density, n, of atoms or molecules (as needed), from the density, Avogadro's number (6.02E23) and the molecular weight (\rho,N_a,W:$$n = \frac{\rho N_a}{W} for a gas with pressure, temperature and the boltzmann constant ...

1

Without analyzing your (or Landau's) argument in depth, I'd like to point you to other alternatives to Laplacian Insufficient Reason. The idea of getting rid of Insufficient Reason is not new and was one of the drivers that motivated the great probability theorist and physicist Edwin T Jaynes, who was unsatisfied with Insufficient Reason unless clear ...

1

For the $h$ factor: (the $N!$ is the Gibbs correction as already explained by others ) Step 1: Semi-classical setting The $h$ factor can actually be understood in a semiclassical way. We are talking about classical ideal gas using classical phase space formulation of classical statistical mechanics, but still from a semi-classical point of view we are not ...

1

You state the second law as : The entropy of the universe always increases. In my college textbook it is stated as : Processes in which the entropy of an isolated system would decrease do not occur, or, in every process taking place in an isolated system, the entropy of the system either increases or remains constant.( F.W.Sears an introduction ...

1

Some of the mathematical aspects of the Liouville operator can be found in the second book by Reed and Simon, in section X.14 (it is not a comprehensive account, but it gives the basic ideas and proofs). In the notes at the end of chapter X, in the part dedicated to section X.14, there is also a quite extensive bibliography that may be useful.

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