The origin of thermal equilibrium

Hyungwon Kim 137 Marina Dr. APT F, Edison, NJ 08817

Cold beer on the table has to be consumed before it becomes warm. It does not require sophisticated knowledge of thermal physics to use this fact in everyday tasks. Even my cat seems to know this phenomenon. If I microwave her wet food too long, she waits until it cools down. As we can see from these examples, we almost instinctively know the phenomenon of thermal equilibrium. However, the physical origin of thermal equilibrium has been one of the most important subjects of statistical physics, and still remains illusive today. In actuality, Ludwig Boltzmann invented the methodology of modern statistical physics while he was studying the mechanism of thermal equilibrium. Let’s take a journey to see how physicists have been trying to unravel the origin of thermal equilibrium. First of all, we should closely look at what thermal equilibrium actually means. Thermal equilibrium implies three features: stationary state, ergodicity, and thermal distribution. Stationary state simply means that thermal equilibrium is a dynamically stable state, and thus the system will remain unchanged unless something extra happens. Ergodicity means that any part of a thermally equilibrated system is at thermal equilibrium. Thermal distribution is the probability distribution that elements in the system should follow. For example, when the initially cold beer “thermalizes”, it will stay thermalized unless we put it back into the refrigerator (stationary state), after which any part of the beer will have the same temperature (ergodicity), and the probability distribution of molecules in the beer will follow a Boltzmann distribution (thermal distribution). All three of these concepts are closely related, but how they are related and how they constitute the notion of thermal equilibrium took a while for physicists to understand. We will review how they are developed in classical physics and then re- formulate them in a quantum mechanical fashion. In the middle of the 19th century, Sir George Stokes wanted to know what the stationary probability distribution of the dynamical theory known at his time was. Therefore, he gave this problem as an exam for the course he was teaching. James Clerk Maxwell, the father of classical electromagnetism, solved the problem in his exam, thereby deriving the Maxwell-Boltzmann distribution. Later, Ludwig Boltzmann used the concept of atoms, which was highly controversial at his time, to derive the kinetic equation of gases, whose only stationary solution is the Maxwell-Boltzmann distribution, and therefore has to be ergodic. Finally, Albert Einstein gave a complete description of Brownian motion based on the concept of atoms, and finished the debate on whether atoms exist. In short, Maxwell predicted the thermal distribution, Boltzmann gave it a theoretical foundation by assuming the existence of atoms, and Einstein validated Boltzmann’s assumption. By combining the works of these historic giants in physics, we are now able to understand thermal equilibrium in a more accurate manner. For example, an initially non- thermal state (cold beer on the table) experiences random collisions governed by Brownian motion and the Boltzmann equation (beer molecules collide with surrounding air molecules and themselves) and finally reaches thermal equilibrium (beer has the same temperature as room temperature). Three important remarks are in order. We need a sort of randomness (or nonlinearity) for this process to happen, and this process, if it happens, will almost always bring the system to thermal equilibrium. Boltzmann argued the latter part by using his famous concept of entropy. Lastly, once it thermalizes, we do not know where it came from. There is no way to find out how cold the warm beer on the table was by just looking at the current state. Although this theory successfully explains how thermal equilibrium happens and why it is so prevalent in our everyday experience, it has a fundamental drawback. This is a classical physics theory! Scientists believe that quantum theory should govern all of the underlying microscopic mechanisms. Arguably, quantum theory is the most thoroughly tested theory of humankind, mainly due to its counter-intuitive consequences, and it has stood against every single challenge. Then, we should be able to describe thermal equilibrium using quantum theory. However, when applying quantum theory to the phenomenon of thermal equilibrium, we immediately see that something goes wrong at a superficial level. Suppose we know everything in detail, including the motion of random noise. Quantum theory postulates that the dynamics of the whole system must be linear. Linear dynamics means that all detailed information of the initial condition remains indefinitely. In other words, cold beer indefinitely remains cold! This is an obvious contradiction to our everyday experience. Since we cannot neglect our experience, we should find out how to resolve this issue within quantum theory. It is mathematically easy to show that linear dynamics cannot bring a system to thermal equilibrium unless it is initially at thermal equilibrium. Then, the information of thermal equilibrium should be encoded in somewhere else, in contrast to classical physics where thermal equilibrium is achieved purely by dynamics. Here we use one of the fundamental concepts in quantum theory, called “entanglement”. Entanglement is quite a bizarre notion that deserves some explanation. If A and B are entangled, A contains information about B. A not only knows about B but cannot separate from the information about B. Therefore, when A and B are entangled, looking at A unavoidably and immediately tells us some information about B in an averaged way. This strange fact was one of the biggest reasons why Einstein did not believe in quantum theory (however, entanglement has been validated by a series of meticulous experiments conducted more than half a century after Einstein’s original doubt in 1935). Surprisingly, entanglement is partially responsible for thermal equilibrium in quantum theory. John von Neumann has shown that almost all, but not all, possible quantum states are highly entangled. This implies that when we pick a random state allowed by quantum theory and look at any part of the state, we would know the average property of the remainder of the state. In our example, the average property is nothing but the temperature. Therefore, a randomly chosen state has a high probability of being at thermal equilibrium. In other words, if we list all possible combinations of a glass of beer on the table, almost all of the combinations exhibit thermal equilibrium. This explains ergodicity, since each sub-region knows the same temperature. This is why we see thermal equilibrium so often. However, we do see non-thermal states in many places. A cold beer is our classic example. von Neumann’s argument cannot explain how a non-thermal state approaches thermal equilibrium. Now we turn our attention to other components of quantum theory. As we stated before, quantum dynamics is linear. From linear algebra, linear dynamics has stationary states, or so-called “eigenvectors” (or eigenstates by physicists’ terminology). Two important facts about these stationary states (eigenstates) are: (1) any arbitrary state can be written as a linear combination of eigenstates and (2) a linear combination of more than two distinct eigenstates is not stationary. Let’s apply these properties to our cold beer example. Quantum mechanically, a glass of cold beer on the table is a very special linear combination of stationary states of a glass of beer plus the table. Due to this peculiar combination, the beer initially appears to be cold. As the quantum dynamics progresses, this non-stationary information of coldness of the beer spreads out to the large surrounding air, and what remains in the beer is the properties of the stationary states. This is the quantum mechanical origin of “equilibration”, where a non-thermal state approaches a stationary state. The last missing word is “thermal”. One of the most widely accepted explanations of “thermal” equilibrium is the Eigenstate Thermalization Hypothesis (ETH). ETH postulates that all stationary states (eigenstates) appear to be thermal if we look at a part of the state. For example, ETH says that all possible stationary states of a glass of beer on the table are at thermal equilibrium. The initial cold beer system is a highly fine-tuned combination of these stationary states. The quantum dynamics exposes the stationary part of the beer, which is thermal equilibrium. So far, ETH has passed a number of numerical tests but still requires more stringent testing. Ultimately, it has to be both mathematically proven and experimentally confirmed. In summary, we have glanced at the history of thermal equilibrium in classical physics and in quantum physics. Classically, dynamics brings an initially non-thermal state to a thermal equilibrium. Quantum mechanically, information about thermal equilibrium is already encoded in the building blocks of the non-thermal initial state. Quantum dynamics reveals the initially hidden property of thermal equilibrium, and as always, the quantum world remains counter-intuitive to our initial perceptions.

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