Maxwell's Demon
Introduction
Maxwell’s demon is a thought experiment created by the physicist James Clerk Maxwell in 1867. The thought experiment is a paradox that seems to violate the second law of thermodynamics.
In this blog post, we will briefly discuss the Maxwell’s demon thought experiment and the second law of thermodynamics.
Maxwell’s Demon
Let’s take a look at the Maxwell’s demon thought experiment without mentioning the second law of thermodynamics first.
In a system of Maxwell’s demon thought experiment consisting of two chambers of gas of the same temperature, a demon controls a small door between two chambers of gas. As individual gas molecules approach the door, the demon quickly opens and closes the door so that fast molecules pass from the left chamber to the right chamber, while slow molecules pass from the right chamber to the left chamber. Because the kinetic temperature of a gas depends on the average kinetic energy of the molecules, the demon’s actions cause the right chamber to warm up while the left chamber cools down.
Second Law of Thermodynamics
The second law of thermodynamics simply states that heat always spontaneously flows from hotter to colder regions of matter, and never the reverse, unless external work is performed on the system. In the Maxwell’s demon thought experiment, assuming the demon can operate the door without performing any work, the experiment result violates the second law of thermodynamics.
Another equivalent statement of the second law of thermodynamics is that the total entropy of an isolated system of spontaneous processes cannot decrease over time. Entropy is a measure of the number of microscopic configurations that correspond to a thermodynamic system in a macroscopic state.
To evaluate the start state and end state of the Maxwell’s demon thought experiment, we could simplify the calculation by assuming there are only four gas molecules in the system, two fast molecules and two slow molecules. In the start state, the are one fast molecule and one slow molecule in each chamber. In the end state, the are two fast molecules in the right chamber and two slow molecules in the left chamber. We will also assume there are only two possible locations for gas molecules in each chamber. Under these assumptions, without having to learn statistical thermodynamics, we can calculate the entropy of the start state and the entropy of the end state using the Shannon entropy formula.
In the start state, because there can only be four possible states of the system,
- fast, slow | fast, slow
- fast, slow | slow, fast
- slow, fast | fast, slow
- slow, fast | slow, fast
where the probability of each state is $\frac{1}{4}$.
The entropy of the start state, according to the Shannon entropy formula, is
$$
\begin{align}
H &= -\sum_{i=1}^{4} \frac{1}{4} \log_2 \frac{1}{4} \\
&= -4 \times \frac{1}{4} \log_2 \frac{1}{4} \\
&= -\log_2 \frac{1}{4} \\
&= 2
\end{align}
$$
In the end state, because there can only be one possible states of the system,
- slow, slow | fast, fast
where the probability of the state is 1.
The entropy of the end state, according to the Shannon entropy formula, is
$$
\begin{align}
H &= -1 \times \log_2 1 \\
&= 0
\end{align}
$$
The entropy of the end state is less than the entropy of the start state, which violates the second law of thermodynamics.
The Paradox of Maxwell’s Demon
A major paradox of the Maxwell’s demon thought experiment is that the demon’s actions do not require any work to be performed on the system. If such demon exists, the second law of thermodynamics would not exist. So far, no such demon has been found in the real world.
References
Maxwell's Demon