Fermi and Bose Statistics

Fermi and Bose statistics, Pressure, Entropy, Free energy, equation of states Presented by: Sadia Liaqat and Janita Abid Presentation outline: • • • • • Fermi and Bose statistics Pressure Entropy Free energy Equation of states Introduction: In classical statistics, particles are treated as distinguishable, and their distribution is explained by Maxwell–Boltzmann statistics. However, at the microscopic level: • particles become indistinguishable • quantum effects become important • particle distribution depends on spin Therefore, we use quantum statistics: Fermi–Dirac statistics Bose–Einstein statistics Continued..... In statistical mechanics, classical statistics is not enough for microscopic particles. Quantum particles behave differently depending on whether they are fermions or bosons, so we use Fermi–Dirac and Bose–Einstein statistics. Why classical statistics was not enough to explain the behaviour of quantum particles? Because microscopic particles such as electrons, photons, atoms, etc. behave according to quantum mechanics, not classical mechanics. Classical statistics fails because it assumes that particles are: Continued..... • distinguishable • independent in occupancy • not restricted by quantum rules But quantum particles are: • indistinguishable • governed by wave nature • controlled by Pauli exclusion principle (for fermions) • able to occupy the same state collectively (for bosons) So we need quantum statistics instead of classical statistics. Quantum statistics includes the study of systems with indistinguishable particles at the microscopic scale, primarily categorized into § Bose-Einstein statistics § Fermi-Dirac statistics What is Fermi and Bose Statistics? Fermi and Bose statistics describe how indistinguishable particles are distributed among available energy states at thermal equilibrium. There are two types: Continued.... 1. Fermi–Dirac Statistics For particles called fermions 2. Bose–Einstein Statistics For particles called bosons These statistics tell us how particles occupy different energy levels. The main difference is whether particles are allowed to share the same quantum state or not. Fermions: Fermions are particles with half-integer spin: 1/2, 3/2, 5/2, ........ Examples : Electron , proton, neutron Fermions obey the Pauli Exclusion Principle: No two identical fermions can occupy the same quantum state simultaneously. So the occupation number is: ni= 0 or 1 Fermions cannot crowd into the same state. That is why electrons fill atomic orbitals one by one. Bosons: Bosons are particles with integer spin: 0,1,2,3,............. Examples: Photon § Bosons do not obey Pauli exclusion principle. So many bosons can occupy the same quantum state: ni = 0,1, 2, 3, ...... Bosons like to stay together in the same state. This is why laser light and Bose–Einstein condensation are possible. ØFermions → like one student per seat ØBosons → like many students sitting on one big carpet Fermi–Dirac Distribution Function Fermi-Dirac distribution: The average number of fermions in an energy state E is: where, E = energy of state μ = chemical potential k = Boltzmann constant T = absolute temperature Bose–Einstein Distribution Function Bose–Einstein Distribution: The average number of bosons in an energy state E: Where, E = energy μ = chemical potential k = Boltzmann constant =1.380649×10−23 J/K T = temperature Pressure in Fermi–Dirac and Bose–Einstein Statistics In quantum statistical mechanics, pressure arises from the collective motion and quantum nature of particles obeying either Fermi–Dirac or Bose–Einstein statistics. For both systems, pressure is not simply due to classical collisions but is deeply connected to the distribution of particles over energy states. It can be derived from the grand canonical ensemble and is related to the grand potential Ω through P=− Ω / V • In a Fermi gas (such as electrons in a metal), pressure exists even at absolute zero due to the Pauli exclusion principle, which prevents fermions from occupying the same quantum state; this gives rise to degeneracy pressure, crucial in systems like white dwarf stars. In contrast, in a Bose gas, particles can occupy the same state, and at low temperatures, they undergo Bose–Einstein condensation, where a macroscopic number of particles occupy the ground state, significantly altering the pressure behavior. At high temperatures, both Fermi and Bose gases approach classical ideal gas behavior, where pressure reduces to P= N kB T/V Entropy in Fermi–Dirac and Bose–Einstein Statistics Entropy in quantum systems measures the degree of uncertainty or distribution of particles among available quantum states. For systems obeying Fermi–Dirac and Bose–Einstein statistics, entropy is calculated using occupation probabilities of energy states. The general expression for entropy involves the occupation number ni of each state and reflects how particles are distributed across energy levels. In a Fermi system, entropy is very low at absolute zero because all states up to the Fermi energy are filled and no rearrangement is possible; hence the system is highly ordered. Entropy formula: Free Energy in Fermi–Dirac and Bose–Einstein Statistics Free energy is a thermodynamic potential that determines the equilibrium and stability of a system. In quantum statistical mechanics, the most relevant form is the grand potential Ω, from which other free energies can be derived. The Helmholtz free energy is given by F=U−TS, where U is internal energy and S is entropy. For Fermi–Dirac and Bose–Einstein systems, free energy depends on how particles occupy energy states according to their respective distribution functions. In a Fermi gas, the free energy reflects the restriction that no two fermions can occupy the same state, leading to a structured filling of energy levels up to the Fermi energy. In a Bose gas, free energy captures the tendency of particles to accumulate in the lowest energy state at low temperatures, leading to Bose–Einstein condensation. Equation of State in Fermi–Dirac and Bose–Einstein Statistics The equation of state provides a relationship between thermodynamic variables such as pressure, volume, temperature, and particle number. In quantum gases, the equation of state is derived using statistical distribution functions. The relationship between P, V, and T for quantum gases deviates from the ideal gas law (PV = N k_B T). For a nonrelativistic quantum gas in three dimensions, the general relationship is: Where U is the internal energy. By expanding the distribution functions, we get the Virial Expansion: Positive sign (+): Represents Fermi gases. The pressure is higher than an ideal gas due to Pauli repulsion. ​Negative sign (-): Represents Bose gases. The pressure is lower than an ideal gas due to the "statistical attraction" of