In the 19th century, James Clerk Maxwell, who developed the modern theory of electromagnetism, introduced Maxwell’s demon and the theory of stochastic thermodynamics was conceived.
Maxwell’s intent is to show how, even though the 2nd law of thermodynamics, that entropy must increase, is always observed in macroscopic systems, it is a statistical law that can be violated at the microscopic level.
Normally cooling systems use changes in pressure and/or volume to change temperature, which increases the entropy in other ways, not violating the 2nd law. But suppose you could do it with some microscopic process without changing volume?
The basic idea behind the thought experiment is that you have some kind of “demon” that can make microscopic adjustments to a system. Maxwell’s thought is that, if you have two systems, A and B, that are in thermal equilibrium, you can cool A while heating B, by having the demon open a little door from A to B for fast moving particles in A and slow moving particles in B that happen to be on a trajectory to hit the door. Eventually, the two systems will be at different temperatures.
In the 1950s, Claude Shannon showed how entropy and information are related. If you have 1 bit to play with, its entropy is proportional to the log of two since the bit has two possible states. (The proportion can be made equality if you multiply by the temperature and Boltzmann’s constant.)
What this means is that Maxwell’s demon is essentially using information (knowledge of the states of individual particles) to decrease the entropy (amount of information) in the system. This process is called feedback control because it involves a measurement and a control given by the demon’s knowledge and its control over the door.
In the last 25 years or so, we have seen the beginnings of technology at the nano-scale being developed that will at long last enable us to create Maxwell’s demon. These 2nd law-violating technologies can exist because the 2nd law of thermodynamics is a statistical law that can be violated for small amounts of time. Indeed, the fluctuation theorem tells us exactly how likely the 2nd law is to be violated.
What this looks like on the nano-scale is that processes usually move forward in time but sometimes appear to move backwards in time. While your broken teacup will never leap up and reassemble itself in your hand, a nano-scale teacup made of only a handful of atoms just might do so before falling over and smashing again.
This new science of stochastic thermodynamics has profound implications for information theory as well because it indicates that it is possible to both measure and control nano-scale particles that are experiencing random, i.e., Brownian, motion just like Maxwell’s demon.
Consider a more modern thought experience called the Szilard engine proposed back in 1929 by Szilard, a student of Albert Einstein. (Einstein and Szilard also invented the Einstein-Szilard refrigerator, a fridge with no moving parts.) The Szilard engine supposes a box containing a single particle moving about inside randomly. Discounting quantum effects, the particle must be in either the right or left side of the box at any given time. A measurement of that detail constitutes extracting a single bit of data from the particle. Left or right.
The Szilard cycle allows us to force the gas to do useful work since we can isolate the particle in one half, then insert some “piston” in the empty half, then remove the partition so that the particle does work on the piston.
The Szilard cycle works by the principle of isothermal expansion. This means an increase of volume under constant temperature. In the case of an ideal gas, this means that the gas is doing work, but this can happen even when the gas is in thermal equilibrium because of the fluctuation theorem, something that the 2nd law of thermodynamics would not normally allow. Because we know the location of the particle on the left or right side of the box, we can allow the particle to expand in the left or right direction. This expansion could lead to cooling technology without moving parts (purely solid state).
Whatever the measure of extraction, this constitutes the first step in constructing Maxwell’s demon. Once we know right or left, that single bit of information will allow us to decide what side to place our energy extraction mechanism (our piston). We know based on the fluctuation theorem the cost of extracting one bit of information.
Now, having that one bit of information, we can turn to the feedback control portion of the demon. In Maxwell’s original idea this was the question: should we let the particle pass through the door or not?
This involves taking our bit of information and now correlating the state of the particle with it, essentially transferring information from our measurement device to the particle.
What the fluctuation theorem for information transfer says is that on average the amount of entropy you produce in a system and the amount of information you transfer to or from it will cancel each other out. Thus, as you gain information you also gain entropy. As you control something else, it gains entropy as well by taking on information from you in the form of control.
Recent experiments have realized Szilard’s thought experiment using a single electron and a voltage gate that controls the electron position and monitored by a single electron transistor.
They can also be realized using long colloidal particles in two optical traps (laser traps). Using electrostatic voltages, the particle can be biased to one or the other trap. Unlike the electron which has an uncertain position, we can detect the particle position inside the traps. The particle is measured and then the traps biased one way or another based on the measurement.
Other realizations have all come out in the last five years, showing the intense interest in 2nd law “violations” and opening the door to nanotechnology based on stochastic thermodynamic principles.
In a dynamical information flow model, we can even now understand how information flows back and forth between two systems and understand the true physical nature of information and how to extract work from systems that we previously thought were just reservoirs of heat energy.
Sagawa, Takahiro, and Masahito Ueda. “Fluctuation theorem with information exchange: Role of correlations in stochastic thermodynamics.” Physical review letters 109.18 (2012): 180602.
Parrondo, Juan MR, Jordan M. Horowitz, and Takahiro Sagawa. “Thermodynamics of information.” Nature physics 11.2 (2015): 131–139.