The applications of statistical mechanics#

Difference between statistical mechanics with classical mechanics, electrodynamics, and quantum mechanics#

Here, we are going to make a comparison between statistical mechanics and other fundamental subjects that we encountered in physics.

Subject

Status of the system

Evolution in time

Classical mechanics

\(\{\textbf{x}_{j}(t)\}, \left\{\frac{d \textbf{x}_{j}(t)}{d t}\right\}\)

\(\mathcal{L\left(\{\textbf{x}_{j}(t)\}, \{\frac{d \textbf{x}_{j}(t)}{d t}\},t\right)}\)

Electrodynamics

\(\textbf{A}(\textbf{x},t), V(\textbf{x},t)\)

Maxwell’s equations

Quantum mechanics

\(\vert \psi(t)\rangle\)

\(i\hbar\frac{\partial \vert \psi(t)\rangle}{\partial t}=\widehat{H}\vert \psi(t)\rangle\)

In the above table, we found the structure of the theory of classical mechanics, electrodynamics, and quantum mechanics all describes the system’s status by some functions and determines the evolution of the system in time through the equation of motion.

Statistical mechanics is different. It is a theory for a large number of degrees of freedom. We don’t have the full information on the status of the system. Therefore, it is impossible and impractical to track the full evolution of our system. To make progress, we need to adjust the approach. Instead of the precise initial information, we focus on the statistical properties of the initial conditions. We will be satisfied if we can make predictions about the system’s statistical properties.

Effective theory beyond the microscopic understanding of thermodynamics#

As we stressed in the previous discussions, the idea of effective theory and emergent phenomena is beyond the connection between microscopic physics and macroscopic physics. Also, since the framework is not tied with microscopic physics, the application of statistical mechanics covers a huge range of physics. Here, we just show some exciting applications and leave our readers to explore the beautiful connection between statistical mechanics and X!