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Statistical mechanics has come a long way from these humble beginnings, but thermodynamics is still an important field in its own right. In this chapter, I will discuss some of the most important results of classical thermodynamics as seen from a modern statistical viewpoint.
Statistical thermodynamics is a theory that uses molecular properties to predict the behavior of macroscopic quantities of compounds. While the origins of statistical thermodynamics predate the development of quantum mechanics, the modern development of statistical thermodynamics assumes that the quantized energy levels associated with a ...
- Blem 6.1
- Blem 6.2
- Blem 6.3
- Blem 6.4
- Blem 6.5
- Blem 6.6
- Blem 6.7
- Blem 6.8
- Blem 6.9
- Blem 6.10
Legend tells us that the inventor of chess asked for \(S=\sum {63}_{z=0}2z\) grains of rice as a wage: one grain on the first square, two on the second, and twice as many on each subsequent square. Compare the sum S for all the squares with the Loschmidt number \(N_\text {L}\approx 6\times 10{23}\). How often can the surface of the Earth be covered...
Justify Stirling’s formula \(n!=(n/{\text {e}})^n\;\sqrt{2\pi n}\) with the help of the equation \(n!=\int ^\infty _0 x^n\;\exp (-x)\;{\text {d}}x\), using a power series expansion of \(n\ln x-x\) about the maximum and also by comparing with \(\ln (n!)\), \(n\ln (n/{\text {e}})\), and \(n\ln (n/{\text {e}})+\frac{1}{2}\ln (2\pi n)\) for \(n=5\), 10...
Draw the binomial distribution \(\rho _z=\left( {\begin{array}{c}Z\\ z\end{array}}\right) \;pz(1-p){Z-z}\) when \(Z=10\) for \(p=0.5\) and \(p=0.1\). Compare this with the associated Gauss distribution (equal to \(\langle z\rangle \) and \(\Delta z\)) and for \(p=0.1\) with the associated Poisson distribution. Note that the Gauss and Poisson distri...
From the binomial distribution for \(Z\gg 1\), derive the Gauss distribution if the probabilities p and \(q=1-p\)are not too small compared to one. Hint: Here it is useful to investigate the properties of the binomial distribution near its maximum and let \(\rho \) depend continuously on z. (8 P)
How high is the probability for z decays in 10 seconds in a radioactive source with an activity of 0.4 Bq? Give in particular the values \(\rho (z)\) for \(z=0\)to 10 with two digits after the decimal point. (6 P)
Which probability distribution \(\{\rho _z\}\) delivers the highest information measure \(I=-\sum _{z=1}Z\rho _z\,\text {lb}\,\rho _z\)? Hint: Note the constraint \(\sum _{z=1}Z\rho _z=1\). How does I change if initially \(Z_1\) states are occupied with equal probability and then \(Z_2
In phase space, every linear harmonic oscillation proceeds along an ellipse. How does the area of this ellipse depend on the energy and oscillation period? By how much do the areas of the ellipses of two oscillators differ when their energies differ by \(\hbar \omega \)? Determine the probability density \(\rho (x)\) for a given oscillation amplitu...
A molecule in a gas travels equal distances l between collisions with other molecules. We assume that the molecules are of the same kind, but always at rest, a useful simplification which does not falsify the result. Here all directions occur with equal probability. Determine the average square of the distance from the initial point after nelastic ...
Does \(\rho (t,\mathbf {r})=\sqrt{4\pi Dt}^{\;-3}\,\exp (-r^2/4Dt)\) solve the diffusion equation \(\partial \rho /\partial t=D\Delta \rho \), and does it obey the initial condition \(\rho (0,\mathbf {r})=\delta (\mathbf {r})\)? What is the time dependence of \(\langle r^2\rangle \)? Compare with Problem 6.8. How do the solutions \(\rho (t,\mathbf ...
Consider N interaction-free molecules each of which is equally probable in any of two equal sections of a container. What is the probability for all N molecules to be in just one of the sections? If each of the possibilities since the existence of the world (\(2\times 10^{10}\)) has occurred corresponding to its probability, how long have 100 molec...
- Albrecht Lindner, Dieter Strauch
- 2018
was not so in the 18 th and early 19th century, when Thermodynamics was developed! For instance, the description of matter as made of atoms and molecules — although an old philosophical idea — in terms of statistical mechanics was yet to be invented. Quantum mechanics and special relativity did not exist. In short, there was no microscopic
This textbook provides comprehensive information on general and statistical thermodynamics. It begins with an introductory statistical mechanics course, deriving all the important formulae meticulously and explicitly, without mathematical shortcuts.
Statistical Thermodynamics. Description: Lecture notes on statistical thermodynamics, calculation of macroscopic thermodynamic results, ideal gas mixtures, ideal liquid mixtures, energy, and entropy changes. Resource Type: Lecture Notes. pdf. 174 kB. Statistical Thermodynamics. Download File. DOWNLOAD.
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in the fields of physics, biology, [1] chemistry, neuroscience ...
