Maxwell’s demon walks into Wall Street

What if Maxwell’s demon knew economics? CQT researchers and their collaborators gain a new perspective on ‘work’ by combining economic theory and thermodynamics.

It is well known that the connection between thermodynamics and information theory runs deep. Now, CQT’s Valerio Scarani and Peter Sidajaya and their collaborators in Japan and the United Kingdom have found value in integrating a third field – economic theory. They publish their findings in Physical Review Letters on 9 November 2023.

Thermodynamics helps to describe how a system evolves from one state to another. A fundamental tenet is the second law of thermodynamics, which states that the entropy or disorder of a system always tends to increase. Thermodynamics links to information theory through this concept of entropy.

Maxwell’s demon, conceived in the 19th Century, made the connection clear. In the thought experiment, an all-knowing demon sorts gaseous molecules between two chambers, such that faster-moving molecules are on one side and slower-moving molecules on another side. The demon thus decreases the entropy of the system. This seems to violate the second law, until you realise that the demon’s information processing takes work.

Researchers now study trends in entropy production using fluctuation relations. These relations consider how fluctuations in a system affect its evolution. They hold in a variety of systems ranging from mechanobiology, such as the folding and unfolding of proteins, to nanotechnology, for example levitating dielectric nanospheres.

Working on Crooks’ fluctuation relation, an equation that describes such processes, the researchers realised they could make a connection with economic theory. They found that a statistical signature in Crooks’ relation could be interpreted as the risk aversion of an agent in ‘expected utility theory’. Co-author Francesco Buscemi has written a short blog post summarising the work.

Valerio says, “For me, what is interesting is that while the mathematical properties of these processes have been studied for many years, this property was not noticed before.”

Economic tools

Widely used in economic science, expected utility theory studies how rational agents make decisions involving uncertainty. One such situation could be a tax game. A tax collector gives a player two options: (1) paying $0 or $100 depending on a random coin toss; and (2) paying a fixed amount of tax between $0 and $100. It turns out that the player’s choice hinges on their risk aversion and the fixed amount set in option 2, known as the ‘certainty equivalent’.

A more risk-averse player chooses option 1 when the certainty equivalent is higher than $50. A more risk-seeking player chooses option 1 when the certainty equivalent is less than $50. The certainty equivalent varies from player to player according to the player’s risk aversion

Integrating economics and thermodynamics

Crooks’ relation considers the entropy produced, or work dissipated, as a system evolves forwards or backwards. It predicts the overall work dissipated by making a statistical comparison between the probability distributions of the two processes. The overall work indicates which direction the system is likely to evolve.

The researchers consider a family of ‘statistical divergences’, relations that quantify the distance between the two probability distributions. This family is indexed by a parameter, r.

To interpret r in thermodynamics, the researchers reframe the work dissipation to be like the tax game. Imagine a player now has a charged battery to carry out a physical process. They could choose either to carry out a physical process that dissipates a stochastic amount of work obeying Crooks’ fluctuation relation, or carry out another process without fluctuations that dissipates a deterministic amount of work.

In this scenario, r becomes like the risk aversion of players in the tax game. Positive values of r indicate a risk-averse player while negative values of r indicate a risk-seeking player. A risk-averse player switches to the stochastic option if the deterministic process dissipates work more than or equal to the certainty equivalent. A risk-seeking player switches to the deterministic process if it dissipates work less than or equal to the certainty equivalent. In this view, the standard calculation of the average entropy production corresponds to a risk-neutral player (r=0).

Based on some of their earlier work linking logic to fluctuation relations, the researchers initially thought that r values might be interpreted in terms of ignorance of the backward process. Valerio says, “We were hoping to validate our intuition about the backward process by saying if you defined it wrongly, you suffer more economically. But it turns out this is not the case.”

So far, the content of this work is entirely within the realm of classical thermodynamics. “The underlying process may be quantum, but all that is needed for our result is that dissipated work obeys the classical Crooks’ fluctuation theorem” says Valerio. “Going fully quantum is our next step”.