When we observe an everyday phenomenon like the heating of water, we usually describe it using concepts like temperature or pressure. This description falls within the realm of classical thermodynamics, which deals with the overall properties of systems without delving into microscopic details.
However, behind every degree that the thermometer rises, there are millions of particles in constant motion.
Statistical thermodynamics emerges to make sense of this invisible dynamic. It uses tools from statistical mechanics to explain how the collective behavior of atoms and molecules gives rise to the properties we perceive on a macroscopic scale.
This discipline allows us not only to better understand thermodynamic laws, but also to predict phenomena that traditional thermodynamics alone cannot explain.
What does statistical thermodynamics study?
Statistical thermodynamics seeks to relate the macroscopic properties of a system (such as temperature, energy, or entropy) to the microscopic characteristics of its component particles. Its starting point is to consider all possible microstates a system can have —that is, all possible configurations of particle positions and energies—and analyze which are most probable.
Through this probabilistic approach, an accurate description of the system's macrostate —that is, the set of observable properties—can be obtained. This type of analysis is essential for understanding, for example, how the velocities of molecules are distributed in a gas or why some materials conduct heat better than others.
Fundamental elements
1. Microstates and macrostates
A single observable state (macrostate) can be compatible with multiple different microscopic configurations.
The entropy of a system, for example, is directly related to the number of possible microstates: the greater the number of compatible configurations, the greater the entropy. This idea was formulated by Ludwig Boltzmann , whose famous principle is summarized in the formula:
where S is the entropy, k is the Boltzmann constant and Ω is the number of microstates compatible with the macrostate.
2. Partition function
A central tool in this discipline is the partition function, represented as Z. This concept is key to the canonical formulation of statistical mechanics and is described in detail in textbooks such as Statistical Physics by Landau and Lifshitz. The partition function allows us to derive quantities such as internal energy, entropy, or pressure from the microstates of the system.
3. Statistical distributions
The type of particles and their physical nature determine which statistical distribution should be used:
- Maxwell-Boltzmann : For non-indistinguishable classical particles.
- Fermi-Dirac : For quantum particles with half-integer spin (fermions), such as electrons.
- Bose-Einstein : For bosons, which can share quantum states.
- These distributions are fundamental for describing gases, low-temperature solids, and complex quantum systems.
Applications
Statistical thermodynamics has transversal applications in several scientific disciplines:
- In materials physics , it allows the calculation of thermal, electronic and structural properties.
- In physical chemistry , it is used to predict equilibria and reaction constants.
- In astrophysics , it helps model dense objects such as white dwarfs and neutron stars.
- In molecular biology , it is used to study processes such as protein folding and the stability of DNA-protein complexes.
- In quantum information and nanotechnology, it provides the theoretical framework for understanding devices operating at scales where quantum effects predominate.
A simple example: speed distribution
In an ideal gas, not all molecules move at the same speed. Some are very fast, others very slow. The distribution of these speeds is described by the Maxwell-Boltzmann law, which predicts how many molecules move at a given speed as a function of temperature.
This distribution helps explain phenomena such as evaporation, diffusion, and even why certain gases can escape from a planet's atmosphere.
Differences between statistical and classical thermodynamics
Thermodynamics, as a fundamental branch of physics, can be approached from two complementary but conceptually distinct perspectives: classical and statistical. While classical thermodynamics relies on macroscopic laws formulated from experimental observation, statistical thermodynamics seeks to explain these laws at a microscopic level, using tools from probability theory and quantum mechanics.
The following table summarizes the main differences between the two approaches, highlighting how each approaches the study of physical systems and what types of phenomena it is capable of describing.
| Aspect | Classical thermodynamics | Statistical thermodynamics |
|---|---|---|
| Perspective | Macroscopic | Microscopic |
| Object of study | Global properties of systems (temperature, pressure, volume, etc.) | Statistical behavior of individual particles (atoms and molecules) |
| Method of analysis | Based on empirical laws derived from observation | Based on probabilistic models and statistical theory |
| Origin of the laws | Axiomatic: laws are accepted as postulates | Deduction from the probability of microstates |
| Concept of entropy | Thermodynamic magnitude defined by Clausius | Measure of the number of microstates compatible with a macrostate: S=klnΩS = k \ln \Omega |
| Prediction of phenomena | Limited to systems in equilibrium or near equilibrium | It allows to explain fluctuations, out-of-equilibrium behavior, and quantum systems |
| Typical applications | Heat engines, refrigeration cycles, macroscopic systems | Ideal gases, solids, liquids, quantum systems, materials physics, biophysics |
| Nature of particles | It does not consider the individual character of the particles | Consider particles and their specific energy states (Fermions, Bosons, etc.) |
| Level of abstraction | Low to medium (direct relationship with physical experiences) | High (requires knowledge of quantum mechanics and statistics) |
| Representative example | Carnot cycle, Ideal gas law | Maxwell-Boltzmann distribution, partition function, Bose-Einstein condensate |