## Demon Ideal Gas

Program DemonIdealGas models an ideal thermometer called the demon. The demon is in thermal equilibrium with the system of interest such as an ideal gas. The energy of the demon is its only property. The demon randomly visits a particle in the system and attempts to change its velocity. If the change decreases the energy, the change is accepted and the system's lost energy goes to the demon. If the velocity change increases the system's energy, the change is accepted only if the demon has enough energy to give to cover the increase in energy of the system.

The goal of the simulation is to explore how the demon's mean energy and probability distribution are related to the temperature.

Problem: The demon and the ideal gas

Program DemonIdealGas simulates a demon that exchanges energy with an ideal gas of $N$ particles in $d$ spatial dimensions. Because the particles do not interact, the only coordinate of interest is the velocity of the particles. The demon chooses a particle at random and changes each component of its velocity by an amount chosen at random between $-\Delta$ and $+\Delta$. For simplicity, we set the initial demon energy $E_d=0$ and the initial velocity of each particle equal to $+v_0 \skew3\hat{x}$, where $v_0 = (2E_0/mN)^{1/2}$. $E_0$ is the desired total energy of the system, and $m$ is the mass of the particles. We choose units such that $m=1$, and the energy and momentum are dimensionless.

1. Run the simulation using the default parameters $N=40$, $E=40$, and $d=3$. Does the average energy of the demon approach a well-defined value after a sufficient number of energy exchanges with the system? (One time unit is equal to $N$ trial changes.)
2. What is $\overline{E}_d$, the average energy of the demon, and $\overline{E}$, the average energy of the system? Compare the values of $\overline{E}_d$ and $\overline{E}/N$.
3. Fix $N=40$ and double the total energy of the system. (Remember that $E_d = 0$ initially.) Compare the values of $\overline{E}_d$ and $\overline{E}/N$. Consider other values of $N \geq 40$ and $E$ and determine the relation between $\overline{E}_d$ and $\overline{E}/N$.(Because there are finite-size effects that are order $1/N$, it is desirable to consider $N \gg 1$. The trade-off is that the simulation takes longer to run.)
4. You have probably learned in other courses that the average energy of an ideal gas in three dimensions is equal to $\overline{E} = \frac{3}{2}NkT$, where $T$ is the temperature of the gas, $N$ is the number of particles, and $k$ is a constant. Our choice of dimensionless variables implies that we have chosen units such that $k=1$. Use this relation for $\overline{E}$ to determine the temperature of the ideal gas in parts parts (b) and (c). Is $\overline{E}_d$ proportional to the temperature of the gas?
5. Suppose that the energy momentum relation of the particles is not $\epsilon = p^2/2m$, but is $\epsilon=cp$, where $c$ is a constant (which we take to be 1). Consider various values of $N$ and $E$ as you did in part (c). Is the dependence of $\overline{E}_d$ on $\overline{E}/N$ the same as you found in part (c)?
6. $\!\!\!\!\!{^*}$ After the demon and the system have reached equilibrium, we can compute the histogram $H(E_d)\Delta E_d$, the number of times that the demon has energy between $E_d$ and $E_d + \Delta E_d$. The bin width $\Delta E_d$ is chosen by the program. The histogram is proportional to the probability $p(E_d)\Delta E$ that the demon has energy between $E_d$ and $E_d + \Delta E$. What do you think is the dependence of $p(E_d)$ on $E_d$? Is the demon more likely to have zero or nonzero energy?
7. $\!\!\!\!\!{^*}$ Verify the exponential form, $p(E_d)=A e^{-\beta E_d}$, where $A$ and $\beta$ are parameters. How does the value of $1/\beta$ compare to the value of $\overline{E}_d$? We will find that the exponential form of $p(E_d)$ is universal, that is, independent of the system with which the demon exchanges energy, and that $1/\beta$ is proportional to the temperature of the system.
8. Discuss why the demon is an ideal thermometer.

## Resources

Problems 1.7 and 4.29 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).

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