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# 1.4 Problems for Chapter 1

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EXERCISES:

1.1 Entropy and irreversibility: (a) Two heat reservoirs are at temperatures and . They are connected by a metal rod that conducts the heat per unit time from 2 to 1. Do an entropy balance to show that this process is irreversible.

(b) Using the specific values , and , calculate the entropy increase per second.

1.2 Reversible, isothermal expansion: Compute the entropy balance for the experiment described (or shown) in the lecture, using estimated values of the necessary experimental parameters.
Discuss the role of the ideal gas assumption: is it necessary / unnecessary / convenient ?

1.3 Maxwell square:
Draw up Maxwell's square and use it to complete the following equations:
(a) ; (b) ;
(c) (in terms of derivatives of and );
(d) .

1.4 Random directions in 2D: Using your favourite random number generator, sample a number of angles  equidistributed in . Interpret and as the components of a randomly oriented velocity vector with .
(a) Draw a histogram of the empirical probability (i.e. event frequency) of . Compare the shape of your to the bottom right histogram in Applet Stadium (see 1.2).
(b) Normalize the histogram such that a sum over all bins equals one. What is the value of ?

1.5 Binomial distribution: Suggest an experiment with given and . Perform the experiment times and draw an empirical frequency diagram; compare with 1.39.

1.6 Density fluctuations in air:
(a) Calculate the mean number of molecules (not discerning between and ) that are to be found at a pressure of in a cube with a side length of the wave length of light ( ). What is the standard deviation of the particle number, both absolutely and relative to the mean particle number? (Air is to be treated as an ideal gas at normal temperature .)
(b) Compute the value of the probability density of the event , i. e. the probability of finding an integer number next to the mean number of particles in the sample volume? (Hint: Don't attempt to evaluate factorials of large numbers, such as appear in the binomial distribution ; rather, use that distribution which resembles when becomes large.)
What is the probability of finding only percent of the mean particle number in the sample volume?

1.7 Multinomial distribution: A volume is divided into equal-sized cells. The particles of an ideal gas may be allotted randomly to the cells.
a) What is the probability of finding in a snapshot of the system the partitioning ? Explain the formula.
b) Demonstrate numerically that the partitioning with the greatest probability is given by . For example, compare the situations , , , , and to the most probable one.
c) (1 bonus point) Prove analytically that is the most probable distribution. Hint: minimize the function under the condition .

1.8 Transformation of a distribution density: Repeat the calculation of Example 3 for the two-dimensional case, i.e. and . Draw the distribution density .

TEST YOUR UNDERSTANDING OF CHAPTER 1:

1.1 Thermodynamic concepts:
- What does the entropy balance tell us about the reversibility/irreversibility of a process? Demonstrate, using a specific example.
- Describe the process of thermal interaction between two bodies. When will the energy flow stop?
- Which thermodynamic potential is suited for the description of isothermal-isochoric systems?

1.2 Model systems:
- Describe 2-3 model systems of statistical mechanics.
- What quantities are needed to completely describe the momentary state of a classical ideal gas of particles?
- What quantities are needed for a complete specification of the state of a quantum ideal gas?

1.3 Statistical concepts:
- Explain the concepts distribution function'' and distribution density''; give two examples.
- What are the moments of a distribution? Give a physically relevant example.

1.4 Equal a priori probability: Explain the concept and its significance for statistical mechanics.   Next: 2. Elements of Kinetic Up: 1. Why is water Previous: 1.3 Fundamentals of Statistics
Franz Vesely
2005-01-25