Fermi–Dirac Statistics

Right. You want a Wikipedia article rewritten. And not just rewritten, but… enhanced. As if a dry recitation of facts wasn't enough. You want it to have… flavor. Fine. Let’s see if we can inject a little life into this corpse of a description. Don’t expect sunshine and rainbows. This is about fermions, not a kindergarten field trip.

Fermi–Dirac statistics: A statistical description for the behavior of fermions

Fermi–Dirac statistics represents a cornerstone in the edifice of quantum statistics, specifically designed to delineate the behavior of systems composed of a multitude of non-interacting, identical particles that, crucially, adhere to the tenets of the Pauli exclusion principle. The direct consequence of this adherence is the characteristic Fermi–Dirac distribution, which meticulously maps the distribution of these particles across available energy states. The genesis of this statistical framework is attributed to the independent insights of Enrico Fermi and Paul Dirac, both arriving at the formulation of this distribution in the year 1926. This statistical approach is an integral component of the broader field of statistical mechanics, and its underpinnings are deeply rooted in the principles of quantum mechanics.

At its core, Fermi–Dirac statistics is invoked when dealing with identical and, by definition, indistinguishable particles possessing half-integer spin – particles colloquially known as fermions – that find themselves within a state of thermodynamic equilibrium. The assumption of negligible interaction between these particles allows for a description centered on individual energy states. The resulting Fermi–Dirac distribution, a direct manifestation of the Pauli exclusion principle, dictates that no two identical fermions can occupy the same quantum state. This constraint, seemingly simple, profoundly influences the macroscopic properties of the system. It's most commonly applied to the ubiquitous electrons, fundamental fermions with a spin of spin 1/2.

It's crucial to recognize its counterpart, Bose–Einstein statistics, which governs the behavior of identical, indistinguishable particles with integer spin (0, 1, 2, and so on), known as bosons. In stark contrast, classical physics employs Maxwell–Boltzmann statistics for identical particles that are treated as distinguishable. The key divergence lies in the occupancy of states: while Bose–Einstein and Maxwell–Boltzmann statistics permit multiple particles in a single state, Fermi–Dirac statistics rigorously forbids it.

Equilibrium thermal distributions

The visual representation of thermal distributions starkly illustrates the fundamental differences between particle types. For fermions (blue line), the distribution shows a sharp cutoff due to the Pauli exclusion principle. Bosons (red line), on the other hand, exhibit a tendency to congregate in lower energy states, a phenomenon crucial for phenomena like superfluidity. Classical particles, obeying Maxwell–Boltzmann statistics (green line), fall somewhere in between. The average occupancy, denoted as $\langle n \rangle$ , is depicted against energy $\epsilon$ relative to the system's chemical potential $\mu$ , with $T$ representing the temperature and $k_B$ the Boltzmann constant. The chemical potential, $\mu$ , is a vital parameter that shifts with temperature and particle density, effectively acting as a gatekeeper for particle distribution.

History: Unraveling the Electron's Mysteries

Prior to the advent of Fermi–Dirac statistics, certain phenomena involving electrons proved stubbornly resistant to explanation, often presenting seemingly contradictory evidence. A classic example is the electronic heat capacity of metals. At room temperature, classical theory predicted that electrons should contribute significantly, yet experimental observations suggested that only a tiny fraction, perhaps a hundredth of the expected number, were actually participating. This discrepancy was a persistent thorn in the side of the prevailing Drude model, the early electronic theory of metals. The model's fundamental flaw lay in treating electrons as equivalent in a classical sense, assuming each contributed to specific heat on the order of the Boltzmann constant, $k_B$ . This assumption, it turned out, was catastrophically wrong.

Another perplexing observation was the near-independence of emission currents generated by high electric fields applied to metals at room temperature. This temperature insensitivity was difficult to reconcile with classical expectations.

The resolution, as it so often does in physics, came with a paradigm shift. Fermi–Dirac statistics, first formally articulated by Enrico Fermi in 1926 and concurrently by Paul Dirac, provided the necessary framework. It’s worth noting that, according to Max Born, Pascual Jordan had independently developed similar statistical concepts in 1925, referring to them as "Pauli statistics," though his work did not achieve the same prompt dissemination. Dirac himself credited Fermi with the initial study, which led him to coin the terms "Fermi statistics" and "fermions."

The impact of this new understanding was immediate and profound. In 1926, Ralph Fowler ingeniously applied Fermi–Dirac statistics to explain the gravitational collapse of stars, leading to the concept of white dwarf stars. By 1927, Arnold Sommerfeld utilized it to refine the free electron model of metals, offering a more accurate picture of electron behavior. The following year, Fowler and Lothar Nordheim extended its application to field electron emission from metals. Fermi–Dirac statistics, therefore, didn't just solve existing problems; it opened entirely new avenues of inquiry and solidified its place as an indispensable tool in physics.

Fermi–Dirac distribution: The Mathematical Heartbeat

For a system of identical fermions in thermodynamic equilibrium, the average number of fermions occupying a specific single-particle state $i$ is precisely quantified by the Fermi–Dirac (F–D) distribution:

${\bar {n}}_{i}={\frac {1}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}+1}}$

Here, $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, $\varepsilon_i$ represents the energy of the single-particle state $i$ , and $\mu$ denotes the total chemical potential. The distribution is normalized by the condition that the sum of average occupancies over all states must equal the total number of particles, $N$ :

$\sum _{i}{\bar {n}}_{i}=N$

This normalization allows us to express the chemical potential $\mu$ as a function of temperature $T$ and particle number $N$ , i.e., $\mu = \mu(T, N)$ . It's important to note that $\mu$ can assume either positive or negative values, a detail that has significant implications.

At the absolute zero of temperature ( $T=0$ ), the chemical potential $\mu$ converges to the Fermi energy plus the potential energy per fermion, assuming a positive spectral density in its vicinity. However, in cases with a spectral gap, such as in semiconductors, the point of symmetry, $\mu$ , is more appropriately termed the Fermi level or, for electrons, the electrochemical potential, and it resides precisely in the middle of that gap.

The Fermi–Dirac distribution is valid under the assumption that the number of fermions in the system is sufficiently large so that the addition of a single fermion has a negligible impact on $\mu$ . A direct consequence of the Pauli exclusion principle, which limits the occupancy of any given state to at most one fermion, is that the average occupancy $\bar{n}_i$ is always strictly between 0 and 1 ( $0 < \bar{n}_i < 1$ ). This ensures that the principle is never violated, even on average.

The variance of the number of particles in a given state $i$ , a measure of the fluctuation around the average, can be derived from the distribution itself:

$V(n_{i})=k_{\text{B}}T{\frac {\partial }{\partial \mu }}{\bar {n}}_{i}={\bar {n}}_{i}(1-{\bar {n}}_{i})$

This variance is particularly significant in understanding transport phenomena. For instance, it plays a role in the Mott relations for electrical conductivity and the thermoelectric coefficient in an electron gas, as the ability of an energy level to contribute to these phenomena is directly proportional to this variance.

Distribution of particles over energy

While the Fermi–Dirac distribution describes occupancy per state, it's often more practical to consider the distribution of particles across energy levels. This is achieved by multiplying the Fermi–Dirac distribution by the degeneracy, $g_i$ , which represents the number of distinct states sharing the same energy $\varepsilon_i$ .

${\bar {n}}(\varepsilon _{i}) = g_{i}{\bar {n}}_{i} = {\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}+1}}$

When the degeneracy $g_i$ is greater than or equal to 2 ( $g_i \geq 2$ ), it becomes possible for the average number of particles at a specific energy $\bar{n}(\varepsilon_i)$ to exceed 1. This is because multiple states with the same energy can be occupied.

In scenarios where energies form a quasi-continuum, characterized by a density of states function $g(\varepsilon)$ (representing the number of states per unit energy range per unit volume), the average number of fermions per unit energy range per unit volume is given by:

${\bar {\mathcal {N}}}(\varepsilon )=g(\varepsilon )F(\varepsilon )$

Here, $F(\varepsilon)$ is known as the Fermi function, which shares the same mathematical form as the Fermi–Dirac distribution:

$F(\varepsilon )={\frac {1}{e^{(\varepsilon -\mu )/k_{\text{B}}T}+1}}$

Consequently, the distribution of particles over energy in this continuous case becomes:

${\bar {\mathcal {N}}}(\varepsilon )={\frac {g(\varepsilon )}{e^{(\varepsilon -\mu )/k_{\text{B}}T}+1}}$

Quantum and classical regimes: Where the rules change

The beauty of Fermi–Dirac statistics lies in its ability to seamlessly transition into the classical Maxwell–Boltzmann distribution under specific limiting conditions, without requiring any arbitrary adjustments. This occurs in two primary scenarios:

Low particle density: When the average occupancy of any given state, $\bar{n}_i$ , is significantly less than 1 ( $\bar{n}_i \ll 1$ ), it implies that $e^{(\varepsilon _{i}-\mu )/k_{\rm {B}}T}+1 \gg 1$ . This simplifies to $e^{(\varepsilon _{i}-\mu )/k_{\rm {B}}T} \gg 1$ . In this regime, the Fermi–Dirac distribution approximates to: $\bar{n}_{i}\approx {\frac {1}{e^{(\varepsilon _{i}-\mu )/k_{\rm {B}}T}}} = {\frac {N}{Z}}e^{-\varepsilon _{i}/k_{\rm {B}}T}$ This is precisely the Maxwell–Boltzmann distribution, where $Z$ is the partition function.
High temperature: At elevated temperatures, particles tend to occupy a much broader range of energy states. Consequently, the occupancy of individual states, particularly those at higher energies where $\varepsilon_i - \mu \gg k_B T$ , becomes very low ( $\bar{n}_i \ll 1$ ). Again, this reduces the Fermi–Dirac distribution to the Maxwell–Boltzmann form.

The "classical regime," where Maxwell–Boltzmann statistics serves as a valid approximation, is fundamentally linked to the Heisenberg uncertainty principle. It prevails when the separation between particles is significantly larger than their de Broglie wavelength. Mathematically, this condition is expressed as:

${\bar {R}}\gg {\bar {\lambda }}\approx {\frac {h}{\sqrt {3mk_{\rm {B}}T}}}$

Here, $\bar{R}$ is the average interparticle separation, $\bar{\lambda}$ is the average de Broglie wavelength, $h$ is the Planck constant, $m$ is the mass of a particle, and $k_B$ and $T$ are the Boltzmann constant and absolute temperature, respectively.

Consider conduction electrons in a typical metal at room temperature ( $T = 300 \, \text{K}$ ). The concentration of these electrons is remarkably high, leading to a small $\bar{R}$ . Simultaneously, their small mass results in a relatively large $\bar{\lambda}$ . The ratio $\bar{R} \approx \bar{\lambda}/25$ clearly indicates that the system is far from the classical regime. Therefore, Fermi–Dirac statistics are not merely a choice but a necessity for accurately describing these electrons.

Another compelling example is the degenerate electron gas found in a white dwarf star. Despite potentially high surface temperatures (around $10,000 \, \text{K}$ ), the extreme density of electrons, coupled with their small mass, ensures that the system remains deep within the quantum realm, necessitating the application of Fermi–Dirac statistics.

Derivations: Unveiling the Mathematical Structure

The elegance of Fermi–Dirac statistics can be appreciated through its derivation using different thermodynamic ensembles.

Grand canonical ensemble

The grand canonical ensemble provides a particularly straightforward path to deriving the Fermi–Dirac distribution, especially for non-interacting fermions. In this framework, the system can exchange both energy and particles with a surrounding reservoir, which maintains a constant temperature $T$ and chemical potential $\mu$ .

Given the non-interacting nature, each single-particle energy level $\varepsilon$ can be viewed as an independent thermodynamic system in contact with the reservoir. The Pauli exclusion principle dictates that each such level can exist in one of two microstates: either empty (energy $E=0$ ) or occupied by a single fermion (energy $E=\varepsilon$ ). This leads to a simple partition function for each single-particle level:

${\mathcal {Z}} = \exp(0(\mu -\varepsilon )/k_{\rm {B}}T) + \exp(1(\mu -\varepsilon )/k_{\rm {B}}T) = 1 + \exp((\mu -\varepsilon )/k_{\rm {B}}T)$

The average number of particles occupying this single-particle level, $\langle N \rangle$ , is then calculated as:

$\langle N\rangle = k_{\rm {B}}T{\frac {1}{\mathcal {Z}}}\left({\frac {\partial {\mathcal {Z}}}{\partial \mu }}\right)_{V,T} = {\frac {1}{\exp((\varepsilon -\mu )/k_{\rm {B}}T)+1}}$

This result, derived for a single level, directly yields the Fermi–Dirac distribution for the entire system.

The variance in particle number, representing thermal fluctuations, is also elegantly obtained:

${\big \langle }(\Delta N)^{2}{\big \rangle } = k_{\rm {B}}T\left({\frac {d\langle N\rangle }{d\mu }}\right)_{V,T} = \langle N\rangle {\big (}1-\langle N\rangle {\big )}$

This quantity, as mentioned, is crucial for understanding transport properties.

Canonical ensemble

Fermi–Dirac statistics can also be derived within the canonical ensemble, which considers a system with a fixed number of particles $N$ and energy $E$ in thermal equilibrium. For a system of $N$ non-interacting fermions, the total energy $E_R$ of a many-particle state $R$ is the sum of individual particle energies $\varepsilon_r$ , weighted by their occupancy numbers $n_r$ . The probability of being in state $R$ is given by the normalized canonical distribution:

$P_{R}={\frac {e^{-\beta E_{R}}}{\displaystyle \sum _{R'}e^{-\beta E_{R'}}}}$

where $\beta = 1/k_B T$ and $e^{-\beta E_{R}}$ is the Boltzmann factor. The average occupancy number for a single-particle state $i$ is then:

${\bar {n}}_{i}=\sum _{R}n_{i}P_{R}$

By meticulously rearranging the summations and employing approximations for large $N$ , particularly relating the partition functions of systems with $N$ and $N-1$ particles ( $Z_i(N)$ and $Z_i(N-1)$ ), and using the thermodynamic relation $\alpha_i \simeq -\mu / k_B T$ , one arrives at the familiar Fermi–Dirac distribution:

${\bar {n}}_{i}={\frac {1}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}+1}}$

The Darwin–Fowler method of mean values also provides a pathway to this distribution, alongside the Maxwell–Boltzmann and Bose–Einstein distributions.

Microcanonical ensemble

A direct analysis using the multiplicities of the system, coupled with the method of Lagrange multipliers, also yields the Fermi–Dirac distribution. This approach involves maximizing the total number of microstates $W$ subject to constraints on the total number of particles $N$ and total energy $E$ . The number of ways to distribute $n_i$ indistinguishable fermions into $g_i$ distinct sublevels (each with energy $\varepsilon_i$ and respecting the Pauli exclusion principle) is given by the binomial coefficient:

$w(n_{i},g_{i})={\frac {g_{i}!}{n_{i}!(g_{i}-n_{i})!}}$

The total number of microstates $W$ is the product of these binomial coefficients over all energy levels. By applying Stirling's approximation to the factorials and taking derivatives, setting them to zero, and solving for the occupation numbers $n_i$ , one obtains:

$n_{i}={\frac {g_{i}}{e^{\alpha +\beta \varepsilon _{i}}+1}}$

Through thermodynamic arguments, it is established that $\beta = 1/k_B T$ and $\alpha = -\mu/k_B T$ . This leads to the probability that a state $i$ is occupied:

${\bar {n}}_{i}={\frac {n_{i}}{g_{i}}}={\frac {1}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}+1}}$

This derivation, while more involved, underscores the robust nature of the Fermi–Dirac distribution across different statistical mechanical frameworks.

Notes

The F–D distribution is a specific instance of a logistic function, also known as a sigmoid function. Its S-shape is characteristic of transitions.
The average occupancy $\bar{n}_i$ also represents the probability that state $i$ is occupied. Given the Pauli exclusion principle, this probability is strictly bounded: $0 < \bar{n}_i < 1$ .
While the term "Fermi–Dirac distribution" is sometimes loosely applied to the distribution of particles over energy levels, this article specifically reserves it for the distribution over states.

Fermi–Dirac Statistics