There are many very good textbooks on statistical physics, and the purpose of this script is solely to document my personal selection of relevant material. Although somewhat debated, this connection again shows that statistical physics is more than statistical mechanics.
Probability in physics
Frequentist approach
First there are some examples for which naive expectations of this kind fail and a more rigorous theory is called for. To address these problems, the concept of probability can be approached with an axiomatic approach.
Axiomatic approach
We see that the action space is closed for the operations of taking complements and counting unions. The most important point is that we must avoid uncountable unions, because this could lead to pathological situations of the nature of the Banach-Tarski paradox (which says that a sphere can be disassembled into points and can then be reassembled into two spheres because the set of real numbers is uncountable).
Continuous distributions and distribution function
Joint, marginal and conditional probabilities
Marginal probability: we are now only interested in the probability for a subset of all variables, e.g. Thenp(A|B) =0.1 and the probability of being infected if the test is positive is surprisingly small.
Expectation and covariance
The more multivariate datasets we can use for this purpose, the better we will be with these estimates and the more accurate our offers will be.
Binomial distribution
The counter is the number of ways we can select features from N without putting them back. We must therefore correct for the number of ways to rearrange the objects, which is the denominator.
Gauss distribution
We note that the peak and mean of the binomial distribution are equal in this limit. The probability of finding x within an interval of one, two, or three standard deviations around µ is.
Poisson distribution
Random walks
Now we tackle its local properties and see that in the end we get essentially the same results again. Ifr =0 (particle is forced to jump),p+q= 1 and we get D= ∆x2/2∆t, which is the basic definition of the diffusion constant.
Computation with random variables
Addition of random variables
You can characterize a probability distribution either by its moments or by its cumulants (except if its moments are not finite, as for the Lorentz distribution, or if they grow too quickly, then you need to know the distribution directly). Therefore, this sum actually has a Gaussian distribution (normal distribution if we also scale the variance to 1).
Information entropy
The same result can also be achieved by using a little trick rather than a Lagrange parameter. If the wall is removed, the gas expands until it fills the entire room (Figure 2.1).
Micro- and macrostates
As with the harmonic crystal, this will be infinitely many states, but they can be counted.
Density of states
This is the equation of an ellipse in the phase space with area A= π a b representing the number of states with energy ≤ E. Therefore we see that the number of classical states is obtained by dividing the volume of the phase space by Planck's constant.
The fundamental postulate
This example suggests counting states in a classical system by dividing phase space into boxes of size h. Second, we expect the system to spread out its presence in phase space as it moves toward equilibrium.
Equilibrium conditions
Again, it may be more intuitive to rewrite the definition of chemical potential in terms of energy rather than entropy. This means that the chemical potential is the energy cost when the number of particles increases (at constantSandV, i.e. no heat flow and constant volume).
Equations of state for ideal gas
Two-state system
It is basically Shannon's formula, S = −∑ipilnpi with i running over the two possible states, ground and excited. Here we see a very general principle of statistical physics: at low T energy is gained (everyone is in the ground state), and at high T entropy is gained (all states are equally populated).
Einstein model for specific heat of a solid
Therefore, we used the geometric sum in the last step to achieve a closed formula. Again, we can ignore the Gaussian integral and only the first term in the Taylor expansion, s, is important.
Entropic elasticity of polymers
Note that we cannot define temperature in the usual way, 1/T = ∂S/∂E, because S(N,L) does not depend on E. For the polymer, in contrast, an increase in temperature causes an increase in the restoring force and thus the system shrinks.
Statistical deviation from average
Foundation of the fundamental postulate
First, the number of occupied microstates does not change, because the volume of the phase space is conserved. Second, the fact that the volume of the phase space is conserved does not mean that its shape does not change.
Free energy
In principle, we can start from the microcanonical ensemble and calculate it as explained above. Thus, the free energy derives directly from the splitting sum and does not need to be passed through the microcanonical ensemble.
Non-interacting systems
Since all particles are identical, we only need to calculate the individual particle partition sum. 2 N kB T= E caloric equation of state We see that we get the same result as for the microcanonical ensemble.
Equipartition theorem
Due to the ideal gas law, the pressure will also drop exponentially on the same scale. 2 ideal gas: f =3N, each particle has three degrees of translation, but there is no interaction potential.
Molecular gases
The internal rotation contributes to the heat capacity of the diatomic molecule as shown in Figure 3.10 (small bump in front of the plateau). Together with the vibration contribution, this results in a specific heat with two rising parts as shown schematically in figure 3.11.
Specific heat of a solid
In the continuum limit, this gives a wave equation ¨q−c2sq00 =0 with wave speed cs= (κa2/m) 1/2 (speed of sound). Fork = π/a (the Brillouin zone boundary), we have a phase difference of π =−1 and the neighbors oscillate against each other.
Black body radiation
Because the earth has a temperature of the order of 300 K, it radiates in the infrared according to Planck (see above). Therefore, they let in radiation from the sun, but absorb radiation from the earth (ozoneO3 absorbs in the UV and therefore protects us from excessive DNA damage).
Probability distribution
Note that we could have derived the same result through information theory by maximizing the Shannon entropy with the two constraints of given average energy and given average particle number.
Grandcanonical potential
Fluctuations
Note that the calculation of the average energy is now a bit more difficult than in the canonical ensemble, because there are two terms related to β.
Ideal gas
Molecular adsorption onto a surface
The number of adsorbed particles first increases linearly with pressure (or equivalently with concentration) and then plateaus.
Chemical reactions
We see from Figure 5.1 that all cases differ in their outcomes for the number of states. In 1995, rubidium atoms were used to realize a 'Bose-Einstein condensate' in nuclear traps at very low temperatures (Tcrit ≈ 10−7K).
Calculating with occupation numbers
The ideal Fermi fluid
The terms used here are the same as those previously used for the Debye solid and the black body radiation. The 'Fermi pressure' in a Fermi liquid at very low temperature is the cause of the incompressibility of matter and is essentially a result of the Pauli principle.
The ideal Bose fluid
However, we treat only the ground state as discrete and treat all excited states as a continuum as before. N0: number of atoms in the ground state, Ne: number of atoms in the excited state.
Classical limit
The Ising model is the most important model in statistical physics, both historically and conceptually. The Ising model below the critical temperature has a first-order phase transition (with jump at M).
The 1d Ising model
This is one of the most important signatures of a critical point: the system becomes correlated over its entire size. In summary, the exact solution of the Ising chain at cero magnetic field shows that it has a critical phase transition at T = 0.
Transfer matrix
We note that solving the Ising model amounts to an eigenvalue problem with λi being the eigenvalues of T . 2 The magnitude of the correlations must diverge, such that infinitely many terms make a non-finite contribution.
Renormalization of the Ising chain
Clearly, this procedure can now be repeated infinitely, defining an RG flow of the model parameters, e.g., K0, K00, etc. (or K0 = K, K1 = K0, K2 = K00, etc.) for the coupling constant. The overall result can be compared with the exact value f = ln(2 cosh(K)) obtained above from the analytical solution of the Ising chain.
Renormalization of the 2d Ising model
This time we have an (unstable, repulsive) fixed point on finite K. Smaller and larger K values flow away from this fixed point. Thus, RG predicts a critical point at finite temperature, consistent with Onsager's exact solution.
The Peierls argument
What is important is that the difference in energy is simply 2Jl, plus the length of the domain wall surrounding this domain. In the second step, we rewrote the sum in terms of the length of the boundary.
The 2d Ising model
The second term is the number of possible paths starting from any point within this area: 4 for the first step and 3 for each additional step (on a simple 2d cubic lattice). From the result of the magnetization (which is the order parameter of the phase transition) the phase diagram can be constructed.
Perturbation theory
Mean field theory for the Ising model
Note that the realFis is concave everywhere, not just at λ=0, so we can use λ=1 without problems. Therefore one power of m cancels and we have m2=3(zK−1). B)3 As above, vanishes at Tc=zJ/kBand which we can write.
Monte Carlo computer simulations of the Ising model
In MC simulations of the Ising model, it is very common to invert entire clusters (e.g. the Swendsen-Wang algorithm) or to implement concepts of RG numerically. Kc for the critical coupling constant of the 3d Ising model (it is about 4.5), which is known to be correct for all given figures due to a detailed analysis of its uncertainty.
Second virial coefficient
To continue, we now expand 4/(Tr6) for small arguments and use the definitions of Gamma functions, which we then evaluate numerically. This can be better understood by considering the free energy of the system, which becomes concave in the unstable region.
Maxwell construction
Both jumps disappear at the critical point, where the isotherm becomes horizontal at the transition. Even more striking, their behavior becomes almost identical at the critical point—large fluctuations make microscopic details irrelevant.
Fluid-solid phase transition
Distribution functions
The "shape factor" describes the interaction between probe and particles and depends on the experiment. Moving 'up' on the manifold indicates 'irreversible' processes, while 'reversible' processes can be thought of as moving sideways.
Variational principles
The equilibrium value of any unrestricted internal parameter Xj is such that the entropy S is maximized for a given value of the total energy E. The equilibrium value of any unrestricted internal parameter Xj is such that the energy E is minimized for a given value of the entropy S.
Euler and Gibbs-Duhem relations
Note that we get the same result as from the microcanonical or canonical ensemble, except that we have a constant of integration that we cannot determine in thermodynamics. We finally note that we could have obtained the same result much more easily by simply integrating the differentials = (1/T)de+ (p/T)dv, but here we wanted to demonstrate the use of the Gibbs-Duhem equation.
Thermodynamic potentials and Legendre transformations
We note that applying the Legendre transform twice brings us back to the original function. We now see that the free energy F is actually the Legendre transformE[T] of energyE from entropySto-temperatureT (alternatively, it can be derived from βF=S[β]).
Maxwell relations
The leads are reduced using the thermodynamic square, the mathematical relationships between the partial leads (see the last part of this chapter), and the Gibbs-Duhem relation for the chemical potential.
Process-dependance of work and heat
However, the way in which energy is divided between heat and mechanical work is not universal and in this sense the system is not conservative. QADB =∆EAB−WADB >0 Heat flows into the system to compensate for this additional work.
Reversible and irreversible processes
The entropy of the piston increases, but the tank loses exactly the same amount of entropy due to heat flow, and the overall entropy is constant. Otherwise∆S > 0 and the process is irreversible as heat flows from the warmer to the cooler system.
Thermodynamic engines
The following thought experiment was proposed for a perpetuum mobile of the second kind and goes back to Maxwell (1867). In this way, the device can produce work from heat, seemingly violating the second law.
Chemical reactions
