Chemical Potential

Chemical potential is the thermodynamic quantity conjugate to a conserved particle number or charge. It answers several closely related questions:

• how does an equilibrium free energy change when one adds particles under specified conditions?
• how strongly does a reservoir favor one number sector over another?
• which multiplier enforces a prescribed mean number in a maximum-entropy state?
• where does the occupation crossover occur for ideal fermions?
• how close may an ideal-boson reservoir parameter approach the lowest mode?

These are not separate definitions. They are different forms of the same conjugate-variable structure. The important qualifications are the ensemble, the variables held fixed, whether particle number is continuous or integer-valued, and the chosen energy zero.

The slogan “energy required to add one particle” is useful only after those qualifications are supplied. In an interacting system, chemical potential is generally a many-body free-energy difference or derivative, not the energy carried by a particular particle.

Definition at a Glance

For a one-component system with internal energy $U(S,V,N)$,

$$dU = T\,dS - P\,dV + \mu\,dN.$$

Therefore

$$\mu = \left( \frac{\partial U}{\partial N} \right)_{S,V}.$$

Legendre transforms give equivalent derivatives under other reversible constraints:

$$\begin{aligned} \mu &= \left( \frac{\partial F}{\partial N} \right)_{T,V}, \\ \mu &= \left( \frac{\partial G}{\partial N} \right)_{T,P}, \\ \mu &= \left( \frac{\partial H_{\mathrm{th}}}{\partial N} \right)_{S,P}. \end{aligned}$$

Here

$$F=U-TS, \qquad G=U-TS+PV, \qquad H_{\mathrm{th}}=U+PV.$$

The symbols beside the derivative are part of the definition. Adding matter at fixed $(S,V)$ is not the same reversible process as adding it at fixed $(T,P)$, even though equilibrium thermodynamics relates the resulting derivatives.

Setting	Role of $\mu$
macroscopic thermodynamics	derivative of a thermodynamic potential with respect to $N$
finite system	addition and removal free-energy differences
constrained entropy maximization	Lagrange multiplier for mean particle number
grand-canonical ensemble	reservoir control in $H-\mu N$
ideal Fermi gas	center of the thermal occupation crossover
ideal Bose gas	parameter bounded above by the lowest mode energy
charged matter	chemical contribution to an electrochemical potential

What “Cost to Add a Particle” Means

At fixed temperature and volume,

$$dF = -S\,dT - P\,dV + \mu\,dN$$

reduces to

$$dF = \mu\,dN.$$

Thus $\mu$ is the reversible Helmholtz-free-energy change per added particle under those conditions. It includes the energetic and entropic consequences of insertion:

$$\mu = \left( \frac{\partial U}{\partial N} \right)_{T,V} - T \left( \frac{\partial S}{\partial N} \right)_{T,V}.$$

It is therefore not generally a bare interaction energy, kinetic energy, orbital energy, or rest energy. The environment may exchange heat and perform work while the particle is added.

At fixed temperature and pressure,

$$dG = \mu\,dN,$$

so $\mu$ is instead the Gibbs-free-energy change per added particle. This is the natural form for phase and chemical equilibrium at fixed $(T,P)$.

The phrase “addition energy” should consequently name its conditions. For a small electronic island or quantum dot, a spectroscopic threshold may also include charging energy, level spacing, electrostatic work, and coupling to gates or leads. It need not equal one isolated one-particle eigenvalue.

Integer Particle Number

Particle number is discrete in a finite system. Let

$$F_N(T,V)$$

be the canonical Helmholtz free energy in the exact $N$-particle sector. Define the removal-side and addition-side chemical potentials by

$$\begin{aligned} \mu_-(N) &= F_N-F_{N-1}, \\ \mu_+(N) &= F_{N+1}-F_N. \end{aligned}$$

These need not agree. Their difference,

$$\Delta_\mu(N) = \mu_+(N)-\mu_-(N) = F_{N+1}+F_{N-1}-2F_N,$$

is a discrete curvature. At zero temperature it becomes an addition–removal or charge-gap diagnostic when $F_N$ is replaced by the ground-state energy $E_0(N)$.

The smooth derivative

$$\mu = \left( \frac{\partial F}{\partial N} \right)_{T,V}$$

emerges after a suitable interpolation or in a thermodynamic limit where changing $N$ by one produces a negligible relative change.

Number-Sector Stability Window

In contact with a particle reservoir, compare the sector grand free energies

$$\Phi_N \equiv F_N-\mu N.$$

The $N$ sector is favored over both neighbors when

$$\Phi_N \leq \Phi_{N-1}, \qquad \Phi_N \leq \Phi_{N+1}.$$

These inequalities are equivalent to

$$\mu_-(N) \leq \mu \leq \mu_+(N).$$

For a finite system, removal and addition thresholds can differ. The $N$ sector minimizes $F_M-\mu M$ against its nearest neighbors when the reservoir parameter lies between $\mu_-(N)$ and $\mu_+(N)$.

If the inequalities are strict and $k_{\mathrm B}T$ is small compared with both distances to the boundaries, the probability is concentrated near $N$. At a boundary, two neighboring sectors become degenerate in $F_M-\mu M$.

This finite interval explains why a gapped system can have no state at energy $\mu$. Inside an incompressible plateau, changing $\mu$ does not change the equilibrium integer particle number until an addition or removal threshold is crossed.

Chemical Potential as a Lagrange Multiplier

Suppose the equilibrium state is inferred by maximizing von Neumann entropy subject to normalization, mean energy, and mean particle number:

$$\operatorname{Tr}\rho=1, \qquad \operatorname{Tr}(\rho H)=U, \qquad \operatorname{Tr}(\rho N)=\overline N.$$

Extremize the dimensionless functional

$$\begin{aligned} \mathcal L[\rho] ={}& - \operatorname{Tr}(\rho\ln\rho) - \alpha \left( \operatorname{Tr}\rho-1 \right) \\ &- \beta \left( \operatorname{Tr}(\rho H)-U \right) + \beta\mu \left( \operatorname{Tr}(\rho N)-\overline N \right). \end{aligned}$$

Variation with respect to $\rho$ gives

$$\ln\rho = -1-\alpha-\beta(H-\mu N),$$

and therefore

$$\rho_{\beta,\mu} = \frac{ e^{-\beta(H-\mu N)} }{ \Xi }, \qquad \Xi = \operatorname{Tr} e^{-\beta(H-\mu N)}.$$

The dimensionless multiplier is $\beta\mu$; $\mu$ itself has dimensions of energy. Its numerical value is fixed by the mean-number constraint:

$$\overline N = \operatorname{Tr} \left( \rho_{\beta,\mu}N \right).$$

This derivation does not make chemical potential arbitrary. The multiplier is determined by the imposed state constraints, just as inverse temperature is determined by the energy constraint.

Maximum Entropy Principle develops the inference argument, independent multipliers, and moment-matching equations in full. The Grand-Canonical Ensemble page owns the complete Fock-space construction and its fluctuation formulas.

Why a Conserved Charge Is Needed

For an ordinary equilibrium chemical potential, the coupled quantity must be conserved by the isolated dynamics:

$$[H,N] = 0.$$

Then energy and number sectors can be labeled consistently, and a reservoir can exchange the conserved quantity with the system.

More generally, if several commuting conserved charges $Q_a$ are constrained, the equilibrium state can take the form

$$\rho \propto \exp \left[ -\beta \left( H-\sum_a\mu_a Q_a \right) \right].$$

An approximately conserved quantity can support an effective chemical potential over an intermediate timescale. Once reactions or relaxation processes freely change that quantity, its independent multiplier is fixed by the remaining equilibrium constraints and often vanishes.

This distinction matters for quasiparticles. Microscopic atom number may be conserved while phonon number is not. A driven photon population may have a useful effective chemical potential even though blackbody photons in complete thermal equilibrium have $\mu_\gamma=0$.

Grand-Canonical Number Weights

Decompose the grand partition function into canonical sectors:

$$\Xi(T,V,\mu) = \sum_N e^{\beta\mu N} Z_N(T,V).$$

Since

$$F_N = -k_{\mathrm B}T\ln Z_N,$$

the sector probability is

$$P(N) = \frac{ e^{-\beta(F_N-\mu N)} }{ \Xi }.$$

The ratio of neighboring probabilities is

$$\frac{P(N+1)}{P(N)} = \exp \left\{ -\beta \left[ F_{N+1}-F_N-\mu \right] \right\}.$$

Therefore:

• if $\mu<\mu_+(N)$, the reservoir suppresses the $N+1$ sector relative to $N$;
• if $\mu>\mu_+(N)$, it favors the $N+1$ sector;
• if $\mu=\mu_+(N)$, the two sectors have equal grand weight before degeneracy factors already contained in $F_N$ are separated out.

For a large smooth system, the dominant sector approximately minimizes

$$F(N)-\mu N.$$

Stationarity gives

$$\left( \frac{\partial F}{\partial N} \right)_{T,V} = \mu.$$

This is the bridge between the reservoir multiplier and the free-energy derivative.

Fugacity and Activity

The fugacity is

$$z \equiv e^{\beta\mu}.$$

Then

$$\Xi = \sum_N z^N Z_N.$$

Because $z>0$ for real $\mu$, it is often convenient in series expansions and ideal-gas calculations. The mean number follows from

$$\overline N = z \left( \frac{\partial\ln\Xi}{\partial z} \right)_{T,V}.$$

In classical solution thermodynamics, an activity $a$ is a dimensionless effective concentration and the chemical potential is commonly written

$$\mu = \mu^\circ(T,P) + k_{\mathrm B}T\ln a.$$

The standard-state term $\mu^\circ$ and the convention used for $a$ must be stated. Fugacity and activity encode interactions and reference choices differently in different subfields; neither should be silently identified with a raw particle density.

Exchange Equilibrium

Consider two subsystems that can exchange energy and particles while the totals are fixed. Their total entropy is

$$S_{\mathrm{tot}} = S_A(U_A,N_A) + S_B(U_B,N_B).$$

Using

$$dS = \frac{1}{T}\,dU - \frac{\mu}{T}\,dN$$

at fixed volume, and

$$dU_B=-dU_A, \qquad dN_B=-dN_A,$$

one obtains

$$\begin{aligned} dS_{\mathrm{tot}} ={}& \left( \frac{1}{T_A} - \frac{1}{T_B} \right)dU_A \\ &- \left( \frac{\mu_A}{T_A} - \frac{\mu_B}{T_B} \right)dN_A. \end{aligned}$$

Independent exchange equilibrium requires

$$T_A=T_B, \qquad \mu_A=\mu_B.$$

The more primitive particle-exchange condition is equality of $\mu/T$ when energy transfer is constrained separately. In ordinary mutual thermal equilibrium, equal temperatures reduce it to equality of chemical potentials.

The phrase “particles flow from high chemical potential to low chemical potential” is an isothermal shorthand. Coupled heat flow, electric work, reaction constraints, and sign conventions for charged carriers can modify the useful transport affinity. Nonequilibrium currents are governed by differences between reservoir parameters and by kinetics, not by equilibrium thermodynamics alone.

Fluctuation and Response

The grand potential is

$$\Omega_{\mathrm G} = -k_{\mathrm B}T\ln\Xi.$$

Its number derivative is

$$\overline N = - \left( \frac{\partial\Omega_{\mathrm G}}{\partial\mu} \right)_{T,V}.$$

Differentiating again gives

$$\left( \frac{\partial\overline N}{\partial\mu} \right)_{T,V} = \beta \operatorname{Var}(N) \geq 0.$$

Thus increasing $\mu$ cannot decrease the grand-canonical mean number under the standard equilibrium assumptions. In a homogeneous phase with density $n=\overline N/V$, the isothermal compressibility can be written

$$\kappa_T = \frac{1}{n^2} \left( \frac{\partial n}{\partial\mu} \right)_T,$$

provided the conventions and other fixed variables are understood.

A plateau of $N$ against $\mu$ at zero temperature signals incompressibility. A large derivative signals enhanced number response and, in the grand-canonical ensemble, large number fluctuations. Fluctuations and Susceptibilities develops this structure across energy, number, and magnetization.

Energy-Zero Convention

For number-conserving matter, shift every one-particle contribution by the same constant $C$. At the many-body level this means

$$H' = H+CN.$$

If simultaneously

$$\mu' = \mu+C,$$

then

$$H'-\mu'N = H-\mu N.$$

The density operator, occupation factors, and grand partition function are unchanged. Consequently, the absolute sign of $\mu$ is generally not meaningful without an energy-zero convention.

This joint shift is different from adding one global constant to the entire Hamiltonian:

$$H' = H+C\mathbb I.$$

The latter multiplies $\Xi$ by $e^{-\beta C}$ and shifts the grand potential, but it leaves the normalized state unchanged without requiring a change in $\mu$.

Useful physical statements involve differences such as

$$\epsilon_i-\mu, \qquad \mu_A-\mu_B, \qquad \mu_+-\mu_-,$$

or a chemical potential relative to a declared band edge, vacuum level, ground-mode energy, or standard state.

Sign in the Grand Hamiltonian

The standard convention is

$$K = H-\mu N, \qquad \rho \propto e^{-\beta K}.$$

Positive $\mu$ then favors sectors with larger $N$ because their grand energy $E-\mu N$ is lower.

Some sources define a source parameter with the opposite sign, absorb $-\mu N$ into the Hamiltonian, or use a hole number rather than an electron number. The formulas can all be consistent if the definitions are propagated. Before comparing two expressions, inspect:

1. whether the exponential contains $H-\mu N$ or $H+\mu N$;
2. whether $N$ counts particles, holes, charge, or a shifted density;
3. whether the one-particle energy zero has been shifted;
4. whether $\mu$ or the dimensionless quantity $\beta\mu$ is being differentiated;
5. whether electrostatic potential energy has already been included in $H$.

In the Hubbard model, for example,

$$K = H_t + U\sum_i n_{i\uparrow}n_{i\downarrow} - \mu\sum_i n_i$$

has particle–hole-symmetric half filling at $\mu=U/2$ under standard bipartite assumptions. Rewriting the interaction as

$$U \sum_i \left( n_{i\uparrow}-\frac12 \right) \left( n_{i\downarrow}-\frac12 \right)$$

shifts the same symmetric point to a shifted chemical potential of zero. Both statements are correct in their respective conventions.

Chemical and Electrochemical Potentials

For a species with charge $q$ in an electrostatic potential $\phi$, one often defines an electrochemical potential

$$\widetilde\mu = \mu_{\mathrm{chem}} + q\phi.$$

Its spatial uniformity, rather than that of the chemical contribution alone, characterizes diffusive equilibrium in an electrostatic field.

For electrons, $q=-e$ with $e>0$, so

$$\widetilde\mu_e = \mu_{\mathrm{chem},e} - e\phi.$$

Voltage, electron energy, conventional current, and electron-number current therefore carry several sign choices. A relation such as

$$\mu_L-\mu_R=eV$$

is a convention that must be paired with a definition of $V$ and current direction. Mesoscopic Transport states those conventions for reservoir-driven electronic devices.

In solids, the terms chemical potential and Fermi level are sometimes used for the same equilibrium electrochemical energy. In spatially varying or nonequilibrium situations, it is safer to state exactly which quantity is meant.

Fermions

For an independent fermionic mode of energy $\epsilon_i$,

$$\overline n_i = \frac{1}{ e^{\beta(\epsilon_i-\mu)}+1 }.$$

At finite temperature,

$$\overline n_i = \frac12 \quad \text{when} \quad \epsilon_i=\mu.$$

The chemical potential is the center of the occupation crossover, not the energy possessed by every fermion. Modes well below $\mu$ are nearly occupied and modes well above $\mu$ are nearly empty.

For an ideal fixed-density Fermi gas,

$$\lim_{T\to0^+}\mu(T) = \epsilon_{\mathrm F},$$

where $\epsilon_{\mathrm F}$ is the zero-temperature Fermi energy. At nonzero temperature, $\mu(T)$ generally shifts to maintain the prescribed density.

Several distinctions matter:

• Fermi energy usually denotes the ideal or ground-state filling edge at $T=0$.
• Chemical potential is a thermodynamic derivative or reservoir parameter and can depend on temperature and interactions.
• Fermi level is used variably; in band and device contexts it often means an equilibrium electrochemical potential.
• Quasiparticle energy is an excitation energy in an effective many-body description, often measured relative to $\mu$.

In an insulator or finite gapped system, $\mu$ may lie anywhere within an interval that leaves the particle number unchanged. No one-particle state need exist at that energy. In an interacting system, addition and removal spectra can have broad continua and different thresholds even though a single equilibrium $\mu$ organizes the thermodynamics.

The occupation-law derivation belongs to Fermi–Dirac Statistics, while the density-dependent low-temperature result belongs to the Ideal Fermi Gas.

Bosons

For an independent bosonic mode,

$$\overline n_i = \frac{1}{ e^{\beta(\epsilon_i-\mu)}-1 }.$$

The geometric sum defining this expression converges only when

$$\mu<\epsilon_i.$$

For a finite ideal system with lowest one-particle energy $\epsilon_0$,

$$\mu<\epsilon_0.$$

If the energy zero is chosen at the lowest mode, then $\epsilon_0=0$ and the same statement becomes $\mu<0$. This negative sign is a convention-dependent consequence of the chosen zero, not a pathology.

At fixed boson number, cooling an ideal gas can drive

$$\mu\to\epsilon_0^-$$

while the lowest mode becomes macroscopically occupied. The finite-system inequality remains strict at nonzero temperature; the treatment of a condensate and the equality limit require thermodynamic-limit care.

It is incorrect to conclude that every boson has zero chemical potential. The relevant distinction is number conservation:

• conserved bosonic atoms generally have a nonzero, state-dependent chemical potential;
• equilibrium photons and phonons ordinarily have zero chemical potential because their numbers can change freely;
• driven or approximately conserved quasiparticle populations may admit an effective chemical potential;
• interacting condensates have a thermodynamic chemical potential that need not obey the ideal one-particle bound relative to the same naive energy zero.

For example, in a weakly interacting condensate the Gross–Pitaevskii stationary-state eigenvalue is the many-body chemical potential associated with adding particles. Repulsive interactions can make it positive relative to the bottom of the noninteracting trap. This does not contradict the ideal-gas convergence condition because the interacting Hamiltonian and excitation structure are different.

The mode distribution and its convergence belong to Bose–Einstein Statistics; condensation criteria belong to the Ideal Bose Gas.

Classical Dilute Limit

When occupations are dilute,

$$e^{\beta(\epsilon-\mu)} \gg 1,$$

both ideal quantum distributions reduce to

$$\overline n(\epsilon) \simeq e^{-\beta(\epsilon-\mu)} = z e^{-\beta\epsilon}.$$

For a homogeneous three-dimensional ideal gas with internal degeneracy $g$,

$$n = \frac{g z}{\lambda_T^3},$$

where

$$\lambda_T = \sqrt{ \frac{2\pi\hbar^2}{m k_{\mathrm B}T} }.$$

Hence

$$\mu = k_{\mathrm B}T \ln \left( \frac{n\lambda_T^3}{g} \right).$$

In the dilute regime $n\lambda_T^3/g\ll1$, this expression is negative when the one-particle continuum threshold is chosen as zero. A different energy zero shifts $\mu$ by the same constant.

Classical Limit of Quantum Statistics develops the Maxwell–Boltzmann limit, the energy-zero qualification on $z\ll1$, and the leading Bose and Fermi corrections.

Multiple Species and Reactions

For several species,

$$dU = T\,dS - P\,dV + \sum_a\mu_a\,dN_a.$$

Each independent conserved number has its own chemical potential. If reactions connect the species, only combinations compatible with the conserved charges remain independent.

Consider a reaction written with stoichiometric coefficients $\nu_a$:

$$\sum_a \nu_a A_a=0,$$

where products and reactants carry opposite signs by convention. An infinitesimal reaction progress $d\xi$ changes

$$dN_a = \nu_a\,d\xi.$$

At fixed $(T,P)$,

$$dG = \sum_a\mu_a\,dN_a = \left( \sum_a\nu_a\mu_a \right)d\xi.$$

Chemical equilibrium therefore requires

$$\sum_a\nu_a\mu_a = 0.$$

For a process that freely creates and destroys one unconstrained excitation species, this condition can reduce to $\mu=0$. For coupled reactions, individual species may have nonzero chemical potentials while the reaction-weighted combination vanishes.

External Potentials and Local Equilibrium

Suppose particles experience an external one-body potential $V_{\mathrm{ext}}(\mathbf r)$. In a local-density approximation, equilibrium is often expressed as

$$\mu_{\mathrm{hom}} \left[ n(\mathbf r),T \right] + V_{\mathrm{ext}}(\mathbf r) = \mu_{\mathrm{global}}.$$

Equivalently, define the local effective chemical potential

$$\mu_{\mathrm{loc}}(\mathbf r) = \mu_{\mathrm{global}} - V_{\mathrm{ext}}(\mathbf r).$$

The density at $\mathbf r$ is approximated by that of a homogeneous system at $\mu_{\mathrm{loc}}(\mathbf r)$ and $T$.

This is an approximation, not a new thermodynamic definition. It requires the external potential and equilibrium density to vary slowly compared with the relevant correlation or microscopic length scales. Near sharp interfaces, critical points, or strongly nonlocal phases, gradient and finite-size effects matter.

Quantum Gases in Traps applies this relation to harmonic Bose and Fermi gases, distinguishes global and local Fermi scales, and states the corresponding semiclassical validity conditions.

Phase Equilibrium

For a one-component material at common temperature and pressure, two phases $\alpha$ and $\beta$ coexist when

$$\mu_\alpha(T,P) = \mu_\beta(T,P).$$

If one phase had the lower chemical potential, transferring matter into it would lower the appropriate Gibbs free energy until another constraint intervened.

For mixtures, phase equilibrium requires equality of each independently exchangeable species chemical potential across phases:

$$\mu_a^{(\alpha)} = \mu_a^{(\beta)}.$$

The full Legendre structure, Gibbs–Duhem relation, and coexistence derivatives belong to Thermodynamic Potentials.

Example: One Fermionic Level

For a spinless fermionic level with occupations $n=0,1$ and energy

$$E_n = n\epsilon,$$

the grand energies are

$$K_0=0, \qquad K_1=\epsilon-\mu.$$

At zero temperature:

• $n=0$ is favored when $\mu<\epsilon$;
• $n=1$ is favored when $\mu>\epsilon$;
• the two sectors are degenerate when $\mu=\epsilon$.

At finite temperature,

$$P(1) = \frac{1}{e^{\beta(\epsilon-\mu)}+1}.$$

This equality between a chemical-potential threshold and one orbital energy is special to a noninteracting single-level problem. Adding interactions or additional particles replaces $\epsilon$ by a many-body free-energy difference.

Example: One Bosonic Level

For a bosonic level with $n=0,1,2,\ldots$,

$$K_n = n(\epsilon-\mu).$$

The grand sum is

$$\Xi_\epsilon = \sum_{n=0}^{\infty} e^{-\beta n(\epsilon-\mu)}.$$

It converges only for

$$\epsilon-\mu>0.$$

As $\mu$ approaches $\epsilon$ from below, sectors with arbitrarily large $n$ cease to be exponentially suppressed. The divergence is therefore a statement about the entire number-sector sum, not merely a denominator manipulation.

Example: Coulomb-Blockaded Island

As a schematic model, let an island have electrostatic energy

$$E_N = E_C(N-n_g)^2,$$

where $n_g$ is a continuously tunable gate offset. Neighboring addition thresholds are

$$\begin{aligned} \mu_+(N) &= E_{N+1}-E_N \\ &= E_C \left[ 2(N-n_g)+1 \right], \\ \mu_-(N) &= E_N-E_{N-1} \\ &= E_C \left[ 2(N-n_g)-1 \right]. \end{aligned}$$

Their separation is

$$\mu_+(N)-\mu_-(N) = 2E_C.$$

At low temperature and weak tunneling, the island retains an integer $N$ across a finite reservoir-chemical-potential interval. At a degeneracy boundary, sequential tunneling can change the charge. Real devices also involve orbital levels, capacitance conventions, lead coupling, and nonequilibrium rates.

A Practical Diagnostic

When chemical potential appears in a calculation, ask:

1. What does $N$ count? Particles, holes, charge, atoms, quasiparticles, or a shifted density?
2. Is it conserved? Exactly, approximately, or not at all?
3. Which ensemble is used? Fixed $N$, fixed mean $N$, or coupled reservoirs?
4. Which potential is differentiated? $U$, $F$, $G$, or a finite-sector free energy?
5. What is held fixed? State the derivative subscripts.
6. Is $N$ discrete? If so, distinguish $\mu_-$ and $\mu_+$.
7. What is the energy zero? Quote $\mu$ relative to a declared reference.
8. Is the particle charged? Separate chemical and electrostatic contributions.
9. Is the system ideal, interacting, or quasiparticle-based? Do not transfer ideal-mode bounds blindly.
10. Is the setting equilibrium? Different reservoirs generally imply no single system-wide chemical potential.

Canonical Boundaries

This page owns:

• the unified interpretation of chemical potential;
• derivative, finite-difference, and Lagrange-multiplier forms;
• finite-system addition and removal thresholds;
• energy-zero and sign-convention guidance;
• the conceptual contrast between fermionic, bosonic, and nonconserved excitations;
• brief treatment of electrochemical, reaction, and local chemical potentials.

Other pages own:

• the complete network of thermodynamic potentials and Maxwell relations: Thermodynamic Potentials;
• the Fock-space trace, sector decomposition, and number fluctuations: Grand-Canonical Ensemble;
• equilibrium fluctuation identities and Kubo–Mori covariance: Fluctuations and Susceptibilities;
• named density susceptibility, compressibility normalizations, and local-versus-global source distinctions: Susceptibilities;
• the dilute fugacity and thermal-wavelength criteria: Classical Limit of Quantum Statistics;
• general constrained-entropy inference and multiplier duality: Maximum Entropy Principle;
• occupation-law derivations: Fermi–Dirac Statistics and Bose–Einstein Statistics;
• density-dependent ideal-gas thermodynamics: Ideal Fermi Gas and Ideal Bose Gas;
• the lattice-specific half-filling convention: Hubbard Model;
• atomic occupation intervals and the Mott-lobe convention: Bose–Hubbard Model;
• currents between reservoirs: Mesoscopic Transport.

Common Mistakes

• Calling $\mu$ the energy of a particular particle.
• Writing “the cost to add a particle” without stating what is held fixed.
• Replacing finite addition and removal differences by one derivative without checking system size.
• Assuming a spectral state must exist at energy $\mu$.
• Treating a negative chemical potential as unphysical without naming the energy zero.
• Forgetting that $H\to H+CN$ requires $\mu\to\mu+C$ to preserve $H-\mu N$.
• Using a chemical potential for a nonconserved excitation number without a timescale or reservoir argument.
• Setting $\mu=0$ for every bosonic species.
• Applying the ideal-boson bound $\mu<\epsilon_0$ unchanged to an interacting condensate.
• Assuming $\mu$ always equals the zero-temperature Fermi energy.
• Confusing chemical potential with electrochemical potential for charged matter.
• Reversing signs after switching from electron number to hole number.
• Comparing Hubbard-model values of $\mu$ without checking whether the interaction was particle–hole centered.
• Inferring current direction from equilibrium $\mu$ values while ignoring temperature differences, charge signs, and kinetics.
• Treating the grand-canonical mean $\overline N$ as though every number measurement returned that noninteger value.

Exercises

1. Derivative identities

Starting from

$$dU = T\,dS-P\,dV+\mu\,dN,$$

derive the chemical-potential derivatives of $F(T,V,N)$, $G(T,P,N)$, and $H_{\mathrm{th}}(S,P,N)$.

Solution

For $F=U-TS$,

$$\begin{aligned} dF &= dU-T\,dS-S\,dT \\ &= -S\,dT-P\,dV+\mu\,dN. \end{aligned}$$

Therefore

$$\mu = \left( \frac{\partial F}{\partial N} \right)_{T,V}.$$

For $G=F+PV$,

$$\begin{aligned} dG &= dF+P\,dV+V\,dP \\ &= -S\,dT+V\,dP+\mu\,dN, \end{aligned}$$

so

$$\mu = \left( \frac{\partial G}{\partial N} \right)_{T,P}.$$

For $H_{\mathrm{th}}=U+PV$,

$$\begin{aligned} dH_{\mathrm{th}} &= dU+P\,dV+V\,dP \\ &= T\,dS+V\,dP+\mu\,dN, \end{aligned}$$

hence

$$\mu = \left( \frac{\partial H_{\mathrm{th}}}{\partial N} \right)_{S,P}.$$

2. Finite-sector stability

Show that the $N$ sector minimizes $F_M-\mu M$ against $M=N\pm1$ precisely when

$$\mu_-(N) \leq \mu \leq \mu_+(N).$$

What happens if $\mu_+(N)<\mu_-(N)$?

Solution

Comparing $N$ with $N-1$ gives

$$F_N-\mu N \leq F_{N-1}-\mu(N-1),$$

or

$$\mu \geq F_N-F_{N-1} = \mu_-(N).$$

Comparing $N$ with $N+1$ gives

$$F_N-\mu N \leq F_{N+1}-\mu(N+1),$$

or

$$\mu \leq F_{N+1}-F_N = \mu_+(N).$$

Both can hold only in the stated interval. If $\mu_+<\mu_-$, no reservoir value makes $N$ stable against both nearest sectors. The discrete free energy is locally nonconvex, so equilibrium instead favors another sector or a mixture represented by the convex envelope in a macroscopic description.

3. Grand-canonical state

Maximize von Neumann entropy subject to fixed normalization, mean energy, and mean particle number. Derive the form of $\rho_{\beta,\mu}$ and identify the dimensionless number multiplier.

Solution

Vary

$$\begin{aligned} \mathcal L ={}& - \operatorname{Tr}(\rho\ln\rho) - \alpha(\operatorname{Tr}\rho-1) \\ &- \beta(\operatorname{Tr}\rho H-U) + \gamma(\operatorname{Tr}\rho N-\overline N). \end{aligned}$$

The first variation is

$$\delta\mathcal L = - \operatorname{Tr} \left[ \delta\rho \left( \ln\rho+1+\alpha+\beta H-\gamma N \right) \right].$$

Stationarity for arbitrary allowed $\delta\rho$ gives

$$\rho \propto e^{-\beta H+\gamma N}.$$

Writing

$$\gamma=\beta\mu$$

produces

$$\rho_{\beta,\mu} = \frac{ e^{-\beta(H-\mu N)} }{ \operatorname{Tr}e^{-\beta(H-\mu N)} }.$$

The dimensionless number multiplier is $\gamma=\beta\mu$.

4. Energy-zero invariance

Let $H'=H+CN$ and $\mu'=\mu+C$. Show that the grand-canonical state and all ideal-mode occupations are unchanged. Contrast this with $H'=H+C\mathbb I$.

Solution

For the joint per-particle shift,

$$H'-\mu'N = H+CN-(\mu+C)N = H-\mu N.$$

The exponential, its trace, and the normalized state are therefore identical. At the mode level,

$$\epsilon_i'-\mu' = (\epsilon_i+C)-(\mu+C) = \epsilon_i-\mu,$$

so Bose–Einstein and Fermi–Dirac occupations are unchanged.

For a global shift,

$$H'=H+C\mathbb I,$$

the Boltzmann operator acquires a scalar factor $e^{-\beta C}$. It cancels in the normalized density operator but multiplies $\Xi$ and shifts $\Omega_{\mathrm G}$ by $C$. No change of $\mu$ is needed.

5. One fermionic mode and one bosonic mode

For one mode of energy $\epsilon$, compare the fermionic and bosonic grand sums. Find their mean occupations and state any restriction on $\mu$.

Solution

For a fermionic mode, $n=0,1$:

$$\Xi_{\mathrm F} = 1+e^{-\beta(\epsilon-\mu)}.$$

Hence

$$\overline n_{\mathrm F} = \frac{ e^{-\beta(\epsilon-\mu)} }{ 1+e^{-\beta(\epsilon-\mu)} } = \frac{1}{ e^{\beta(\epsilon-\mu)}+1 }.$$

The finite two-term sum converges for every finite real $\mu$.

For a bosonic mode, $n=0,1,2,\ldots$:

$$\Xi_{\mathrm B} = \sum_{n=0}^{\infty} e^{-\beta n(\epsilon-\mu)} = \frac{1}{ 1-e^{-\beta(\epsilon-\mu)} },$$

which requires $\mu<\epsilon$. Then

$$\overline n_{\mathrm B} = \frac{1}{ e^{\beta(\epsilon-\mu)}-1 }.$$

The different restriction comes from unbounded bosonic occupation, not from a sign convention alone.

6. Particle exchange

Two systems at equal temperature $T$ exchange particles while total particle number is fixed. Let $dN_A=-dN_B$. Show that entropy increases when particles move in the direction selected by the chemical-potential difference, and find the equilibrium condition.

Solution

At fixed energy and volume for the particle-transfer step,

$$dS_i = - \frac{\mu_i}{T}\,dN_i.$$

Therefore

$$\begin{aligned} dS_{\mathrm{tot}} &= - \frac{\mu_A}{T}\,dN_A - \frac{\mu_B}{T}\,dN_B \\ &= - \frac{\mu_A-\mu_B}{T}\,dN_A. \end{aligned}$$

If $\mu_A>\mu_B$, entropy increases for $dN_A<0$: particles leave $A$ and enter $B$ under the stated isothermal, energetically decoupled conditions. Equilibrium requires

$$\mu_A=\mu_B.$$

If energy and particle exchange are coupled, the full entropy differential contains both temperature and chemical-potential affinities.

References

• H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley (1985), especially the treatment of generalized forces, chemical potential, and equilibrium.
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth–Heinemann (1980), chapters on thermodynamic variables and the grand-canonical distribution.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 4th ed., Academic Press (2021), chapters on ensembles and ideal quantum gases.
• M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007), chapters on the grand-canonical ensemble and quantum statistics.
• F. Reif, Fundamentals of Statistical and Thermal Physics, Waveland Press (2009 reissue), sections on diffusive equilibrium and chemical potential.
• C. Kittel and H. Kroemer, Thermal Physics, 2nd ed., W. H. Freeman (1980), chapters on chemical potential and ideal gases.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003 reissue), chapters on quantum gases and many-body thermodynamics.
• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Brooks/Cole (1976), discussions of electron chemical potential, Fermi energy, and equilibrium in solids.
• J. W. Gibbs, Elementary Principles in Statistical Mechanics, Yale University Press (1902), the foundational ensemble treatment.