Scientists are working out the maths of ideal gas expansion in unprecedented detail.
BY DEBDUTTA PAUL
Physicists at ICTS-TIFR are demonstrating afresh how a century-old idea can explain the arrow of time.
Consider the process of mixing milk with coffee. According to the laws of physics, the total energy of the system and its environment remains constant during the entire process. But that holds even for the reverse process: separating milk from coffee. The microscopic phenomena governing the two processes conserve the total energy, but we never spontaneously witness coffee unmixing from milk.
The preference for mixing over unmixing defines the arrow of time.
What creates this preference, even though most microscopic laws of physics apply equally backwards as forwards, is a long-standing puzzle.
The second law of thermodynamics explains the arrow of time via a quantity that physicists call ‘entropy’. According to the second law, entropy increases in any natural process. Ludwig Boltzmann, one of the founders of statistical mechanics, explained this by studying entropy via microscopic phenomena.
Scientists at ICTS-TIFR have demonstrated afresh how entropy increases when an ideal gas expands. Abhishek Dhar, one of the authors of the studies, says that an ideal gas is a gas whose molecules do not interact with each other. Physicists often study simple systems such as ideal gases to understand complex ideas like entropy.
Macrostates, microstates, and Boltzmann’s formula
Physicists describe the gas’s microscopic details via what they call its ‘microstate’. “In an ideal gas, the positions and velocities of all its molecules specify its microstate,” said Anupam Kundu, a co-author of the studies. On the other hand, the physical quantities that describe the gas’s observable properties, like its density and temperature, are examples of ‘macrostates’.
The gas molecules can have many different microstates for a particular macrostate. For example, consider a glass chamber with a partition separating it into two halves. One half contains an ideal gas, while the other half is empty.
Although real gases contain many molecules, below is a cartoon representation of an ideal gas containing only eight molecules. Each half of the glass chamber is divided into ten artificial divisions, represented by green dotted lines. The macrostate has high density on the left half and zero density on the right.
These densities remain unchanged if some molecules move, but the microstate has changed.
There are 45 such microstates corresponding to the same microstate, of which the above are only two examples.
While it is difficult to observe the microstates of real gases directly, physicists can derive insights from them.
The arrow of time for an ideal gas expansion
The researchers at ICTS-TIFR and their colleagues have used Boltzmann’s ideas to demonstrate how the arrow of time emerges in this simple scenario. The team calculated the Boltzmann’s entropy of an ideal gas undergoing expansion using Boltzmann’s formula.
A system is in equilibrium if its macrostates do not evolve with time. Boltzmann’s prescription holds for an isolated system in equilibrium and for one out of equilibrium.
Initially, the ideal gas is in equilibrium. If the partition is suddenly removed, the gas from one half expands and fills the entire chamber. The ideal gas again reaches an equilibrium.
As the gas expands, its macrostates may evolve. For example, since the same number of molecules occupy a larger volume, the gas’s density decreases on the left half but increases on the right.
Below is one such microstate containing seven molecules on the left half and one on the right.
This macrostate corresponds to 1200 such microstates.
According to Boltzmann, the evolving macrostates are such that the number of microstates corresponding to the same macrostate increases. The idea becomes clear as we track the evolution of the gas spreading out in the cartoon scenario.
Now, there are six molecules on the left half and two on the right. The number of such microstates for this macrostate is 9450. Similarly, when the density has reached close to the final equilibrium, with five molecules on the left and three on the right, the number of possible microstates increases to 30240. Below is one example.
The final equilibrium has, on average, four molecules on the left and four on the right. The gas density is the same on the left and the right, and 44100 microstates are possible for this equilibrium.
This number — 44100 — is much larger than the number of microstates corresponding to the initial macrostate with high density on the left half and zero density on the right — 45. The significant increase is evident with only eight molecules. In a real gas containing more than 100,000,000,000,000,000,000,000 molecules, the difference between the number of microstates corresponding to the initial and final equilibria is even more striking.
Ludwig Boltzmann prescribed a formula for a system’s entropy in terms of its microstates. The formula relates the entropy to the number of microstates in the equilibrium macrostate.
Since the final equilibrium macrostate has a higher number of microstates, its entropy is higher. Entropy increases because the gas consists of a large number of constituent molecules. Thus, the arrow of time is not related to the details of the gas’s expansion.
Boltzmann’s view also applies to all other systems we observe around us, which consist of many particles. Thus, the second law of thermodynamics results from the large number of constituents that make up these systems.
The simpler, the better
Abhishek and his colleagues have demonstrated that there is no need to invoke complex ideas like ‘ergodicity’ and ‘chaos’, which other researchers have earlier used to explain the increase in entropy. These ideas are related to the details of the system’s evolution, said Abhishek.
The researchers explained that Botlzmann’s idea of microstates is enough to explain the entropy growth for an ideal gas’ free expansion. In this case, the complex ideas of ergodicity and chaos are not even applicable. “We explicitly constructed the Boltzmann entropy in a specific non-equilibrium situation and showed that it increases with time,” said Abhishek.
Now, the researchers are studying the effect of the molecules’ interactions.
The team’s work has been published in various journals over the years. To know more, read the following papers:
- Saurav Pandey, Junaid Majeed Bhat, Abhishek Dhar, Sheldon Goldstein, David A. Huse, Manas Kulkarni, Anupam Kundu & Joel L. Lebowitz: Boltzmann Entropy of a Freely Expanding Quantum Ideal Gas
- Subhadip Chakraborti, Abhishek Dhar & Anupam Kundu: Boltzmann’s Entropy During Free Expansion of an Interacting Gas
- Subhadip Chakraborti, Abhishek Dhar, Sheldon Goldstein, Anupam Kundu and Joel L Lebowitz: Entropy growth during free expansion of an ideal gas
The author thanks Professors Abhishek Dhar and Anupam Kundu for discussions.