Statistical mechanics¶
Statistical mechanics is a fundamental branch of physics that bridges the gap between the microscopic world of individual atoms and molecules and the macroscopic bulk properties of materials that we observe in everyday life. It provides a framework for understanding how the collective behavior of vast numbers of particles gives rise to the thermodynamic properties of systems at the macroscopic scale.
At its core, statistical mechanics is based on the idea that the properties of a system can be described by the statistical distribution of its constituent particles. By applying probability theory and statistical methods, statistical mechanics allows us to predict and explain various phenomena, such as the behavior of gases, liquids, and solids, as well as more complex systems like plasmas and quantum fluids.
Microscopic states¶
In statistical mechanics, a microscopic state refers to a specific, distinguishable configuration of a system. For our cases, we are concerned primarily with atomistic systems dictated by enormous amount of quantum states. In order to calculate a thermodynamic property such as energy, pressure, or volume we need to account for all possible quantum states. The collection of all possible system states that replicates macroscopic properties is called an ensemble, \(\mathcal{A}\). In other words, the total number of ways you could atomistically arrange particles without changing the number of particles (\(N\)), volume (\(V\)), and energy (\(E\)).
This can be a little confusing, so instead let's consider coin flips. Envision a system that contains exactly three coins that can either be heads or tails. A macrostate is a macroscopic description of our system; for example, one macrostate could be having all three coins show heads. A microstate is a single possible configuration we could arrange our system's components without changing the macrostate. In our case, we only have one way we could arrange our three coins for either all heads or all tails—one one microstate is valid.
What if we defined our macrostate to have two heads and one tails? Well, there would be three possible ways to arrange our coins to achieve this macrostate (see Figure 1).
Figure 1
All possible macrostates (in boxes) and constituent microstates (i.e., coin arrangements) of three coins. This assumes the coins are indistinguishable meaning we cannot tell two coins that show heads apart besides their position.
For example, in a gas, a microscopic state would be defined by the precise positions and velocities of all the individual molecules at a given instant.
It's important to note that a system can have an enormous number of possible microscopic states. To illustrate this, consider a simple system of just 10 particles, each of which can be in one of two possible states (e.g., spin up or spin down). The total number of microscopic configurations for this system would be 2^10 = 1,024. As the number of particles increases, the number of possible microscopic states grows exponentially. For instance, a system of 100 particles with two possible states each would have 2^100 ≈ 10^30 microscopic configurations.
Despite the vast number of microscopic states, many of them can result in the same macroscopic state. For example, consider a gas in a container. The individual molecules can have countless different arrangements of positions and velocities, but as long as the average kinetic energy (related to temperature) and the average force per unit area on the container walls (pressure) remain the same, the gas will be in the same macroscopic state.
Macroscopic properties¶
Macroscopic properties are observable characteristics of a system that emerge from the collective behavior of its microscopic components. Examples of macroscopic properties include temperature, pressure, volume, magnetization, and electrical conductivity.
These properties are typically averages or aggregate outcomes of the microscopic behaviors. For instance, temperature is a measure of the average kinetic energy of the particles in a system. Pressure is the result of the average force exerted by the particles on the walls of a container per unit area.
Macroscopic properties are often associated with the concept of thermodynamic equilibrium, which is a state of balance where the macroscopic properties remain constant over time. In equilibrium, the system's properties are well-defined and can be described by the laws of thermodynamics. The statistical nature of macroscopic properties becomes apparent when we consider that they arise from the average behavior of a large number of microscopic components.
Connecting microscopic to macroscopic¶
The connection between the microscopic and macroscopic descriptions of a system is one of the key insights of statistical mechanics. The macroscopic properties we observe emerge from the collective behavior of the microscopic components through a statistical process.
To understand this connection, consider the example of pressure in a gas. At the microscopic level, pressure arises from the collisions of individual gas molecules with the walls of the container. Each collision imparts a small force on the wall. However, when we have a large number of molecules (on the order of Avogadro's number, ~6.02 × 10^23), the average force per unit area from these collisions becomes the macroscopic pressure we measure.
Similarly, the temperature of a system is related to the average kinetic energy of its particles. As the particles move and collide with each other, they exchange energy, leading to a distribution of kinetic energies. The temperature is proportional to the average of this distribution.
The link between microscopic and macroscopic properties is established through statistical ensembles, which represent the collection of all possible microscopic states of a system consistent with its macroscopic constraints. By applying statistical methods to these ensembles, we can derive the macroscopic properties and their relationships to the microscopic descriptions.