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Statistical physics: microstates to thermodynamics

Derive equilibrium ensembles, free energies, ideal gases, fluctuations and simple interacting models.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Entropy and canonical probabilities

Definitions & inputs. Ω micro state count,pi probability,β1/(kBT),Ei micro state energy,Z partition function.

  1. Equal probability counting and Gibbs entropy connect microscopic uncertainty to entropy.

    S=kBln⁡Ω,S=−kB∑ipiln⁡piS=k_B\ln\Omega,\quad S=-k_B\sum_i p_i\ln p_i

  2. Maximize entropy with normalization and mean energy constraints.

    δ[S−α∑ipi−βkB∑ipiEi]=0\delta[S-\alpha\sum_i p_i-\beta k_B\sum_i p_iE_i]=0

  3. Lagrange multipliers give Bolt z mann weights and the normalize r.

    pi=e−βEi/Z,Z=∑ie−βEip_i=e^{-\beta E_i}/Z,\quad Z=\sum_i e^{-\beta E_i}

Interpretation. βis fixed by the bath;degenerate energy levels contribute their multiplicity to Z.

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2. Thermodynamic derivatives

Definitions & inputs. F Helmholtz free energy,U mean internal energy,P pressure,CV heat capacity.

  1. Partition function summarizes equilibrium state counting.

    F=−kBTln⁡Z,U=−∂βln⁡ZF=-k_BT\ln Z,\quad U=-\partial_\beta\ln Z

  2. Differentiate the appropriate potential to obtain observable s.

    P=kBT(∂Vln⁡Z)T,N,S=(U−F)/TP=k_BT(\partial_V\ln Z)_{T,N},\quad S=(U-F)/T

  3. Asecondβderivative connects energy fluctuations to response.

    Var⁡(E)=kBT2CV\operatorname{Var}(E)=k_BT^2C_V

Interpretation. A partition function must include consistent energy zeros and quantum state-counting normalization.

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3. Classical ideal gas

Definitions & inputs. λth=h/√(2πmkBT),N particle count,Vvolume.

  1. Independent one-particle integrals factorize;N!removes classical over counting.

    ZN=1N!(V/λth3)NZ_N=\frac1{N!}(V/\lambda_{th}^3)^N

  2. Volume and temperature derivatives yield the ideal gas law and translational energy.

    P=NkBT/V,U=32NkBTP=Nk_BT/V,\quad U=\tfrac32Nk_BT

  3. Equipartition assigns kB T/2to each quadratic velocity component.

    vrms=3kBT/mv_{rms}=\sqrt{3k_BT/m}

Interpretation. Internal rotation,vibration,interactions and quantum degeneracy add corrections.

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4. Interactions, correlations and criticality

Definitions & inputs. sij=±1 spins,J interaction energy,H external field energy,g(r) pair distribution.

  1. Interactions favor correlations rather than independent Bolt z mann factors.

    HIsing=−J∑⟨i,j⟩sisj−h∑isiH_{Ising}=-J\sum_{\langle i,j\rangle}s_is_j-h\sum_i s_i

  2. Pair forces correct the ideal gas pressure through structure.

    P=nkBT−2πn23∫0∞r3u′(r)g(r) drP=nk_BT-\frac{2\pi n^2}{3}\int_0^\infty r^3u'(r)g(r)\,dr

  3. Grand-canonical number fluctuations measure a response function.

    ⟨(ΔN)2⟩=kBT(∂N/∂μ)T,V\langle(\Delta N)^2\rangle=k_BT(\partial N/\partial\mu)_{T,V}

Interpretation. Critical phenomena need correlation lengths and finite-size analysis;mean field and Monte Carlo methods have different errors.

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Graphical worked example

Two nondegenerate classical canonical states with energies zero and Δ. The Fermi-like algebra does not make this a Fermi gas. X axis: Energy gap / kBT (dimensionless). Y axis: Two-level excited population (dimensionless).
Two nondegenerate classical canonical states with energies zero and Δ. The Fermi-like algebra does not make this a Fermi gas. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Entropy count

Definitions & inputs. Ω8.

  1. Choose the governing model and isolate the requested quantity.

    S/kB=ln⁡ΩS/k_B=\ln\Omega

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ln⁡8\ln8

  3. Evaluate the expression; the result uses the units shown.

    Result=2.079442 \mathrm{Result}=2.079442\ {}

Interpretation. Equal probability micro states.

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Example 02. Binary entropy

Definitions & inputs. p.5.

  1. Choose the governing model and isolate the requested quantity.

    S/kB=−pln⁡p−(1−p)ln⁡(1−p)S/k_B=-p\ln p-(1-p)\ln(1-p)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −ln⁡(.5)-\ln(.5)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6931472 \mathrm{Result}=0.6931472\ {}

Interpretation. Natural-log entropy.

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Example 03. Thermal energy

Definitions & inputs. T300K,kB1.380649×10⁻²³SI.

  1. Choose the governing model and isolate the requested quantity.

    kBTk_BT

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.380649×10−23(300)1.380649\times10^{-23}(300)

  3. Evaluate the expression; the result uses the units shown.

    Result=4.141947×10−21 J\mathrm{Result}=4.141947\times10^{-21}\ \mathrm J

Interpretation. Per-particle energy scale.

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Example 04. Boltzmann ratio

Definitions & inputs. Gap2kBT,equal degeneracy.

  1. Choose the governing model and isolate the requested quantity.

    p1/p0=e−2p_1/p_0=e^{-2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    e−2e^{-2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1353353 \mathrm{Result}=0.1353353\ {}

Interpretation. Two state weight ratio.

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Example 05. Two-level partition

Definitions & inputs. Levels0andΔ,kBT=Δ.

  1. Choose the governing model and isolate the requested quantity.

    Z=1+e−1Z=1+e^{-1}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1+e−11+e^{-1}

  3. Evaluate the expression; the result uses the units shown.

    Result=1.367879 \mathrm{Result}=1.367879\ {}

Interpretation. Non degenerate levels.

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Example 06. Excited population

Definitions & inputs. Same levels.

  1. Choose the governing model and isolate the requested quantity.

    p1=e−1/(1+e−1)p_1=e^{-1}/(1+e^{-1})

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/(e+1)1/(e+1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2689414 \mathrm{Result}=0.2689414\ {}

Interpretation. Normalized probability.

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Example 07. Mean energy

Definitions & inputs. Δ1eV,kBT1eV.

  1. Choose the governing model and isolate the requested quantity.

    U=Δ/(e+1)U=\Delta/(e+1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/(e+1)1/(e+1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2689414 eV\mathrm{Result}=0.2689414\ \mathrm{eV}

Interpretation. Two state canonical mean.

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Example 08. Two-level heat capacity

Definitions & inputs. xΔ/kBT=1.

  1. Choose the governing model and isolate the requested quantity.

    C/kB=x2ex/(1+ex)2C/k_B=x^2e^x/(1+e^x)^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    e/(1+e)2e/(1+e)^2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1966119 \mathrm{Result}=0.1966119\ {}

Interpretation. Positive Sch ott ky response.

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Example 09. Free energy

Definitions & inputs. kBT1eV,Z1+exp−1.

  1. Choose the governing model and isolate the requested quantity.

    F=−kBTln⁡ZF=-k_BT\ln Z

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −ln⁡(1+e−1)-\ln(1+e^{-1})

  3. Evaluate the expression; the result uses the units shown.

    Result=−0.3132617 eV\mathrm{Result}=-0.3132617\ \mathrm{eV}

Interpretation. Chosen ground energy zero.

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Example 10. Ideal-gas pressure

Definitions & inputs. n2.5×10²⁵/m³,T300K.

  1. Choose the governing model and isolate the requested quantity.

    P=nkBTP=nk_BT

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2.5×1025(1.380649×10−23)(300)2.5\times10^{25}(1.380649\times10^{-23})(300)

  3. Evaluate the expression; the result uses the units shown.

    Result=103548.7 Pa\mathrm{Result}=103548.7\ \mathrm{Pa}

Interpretation. Dilute classical gas.

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Example 11. One-mole energy

Definitions & inputs. T300K,R8.314462618.

  1. Choose the governing model and isolate the requested quantity.

    U=3RT/2U=3RT/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.5(8.314462618)(300)1.5(8.314462618)(300)

  3. Evaluate the expression; the result uses the units shown.

    Result=3741.508 J/mol\mathrm{Result}=3741.508\ \mathrm{J/mol}

Interpretation. Monatomic translational energy.

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Example 12. Molar CV

Definitions & inputs. Monatomic ideal gas.

  1. Choose the governing model and isolate the requested quantity.

    CV=3R/2C_V=3R/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.5(8.314462618)1.5(8.314462618)

  3. Evaluate the expression; the result uses the units shown.

    Result=12.47169 J/(mol K)\mathrm{Result}=12.47169\ \mathrm{J/(mol\,K)}

Interpretation. Internal excitation s absent.

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Example 13. Molar CP

Definitions & inputs. Samegas.

  1. Choose the governing model and isolate the requested quantity.

    CP=CV+RC_P=C_V+R

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2.5(8.314462618)2.5(8.314462618)

  3. Evaluate the expression; the result uses the units shown.

    Result=20.78616 J/(mol K)\mathrm{Result}=20.78616\ \mathrm{J/(mol\,K)}

Interpretation. Ideal-gas identity.

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Example 14. Heat-capacity ratio

Definitions & inputs. Monatomic gas.

  1. Choose the governing model and isolate the requested quantity.

    γ=CP/CV\gamma=C_P/C_V

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    5/35/3

  3. Evaluate the expression; the result uses the units shown.

    Result=1.666667 \mathrm{Result}=1.666667\ {}

Interpretation. Classical translation only.

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Example 15. RMS speed

Definitions & inputs. m4.65×10⁻²⁶kg,T300K.

  1. Choose the governing model and isolate the requested quantity.

    vrms=3kBT/mv_{rms}=\sqrt{3k_BT/m}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    3(1.380649×10−23)(300)/(4.65×10−26)\sqrt{3(1.380649\times10^{-23})(300)/(4.65\times10^{-26})}

  3. Evaluate the expression; the result uses the units shown.

    Result=516.9356 m/s\mathrm{Result}=516.9356\ \mathrm{m/s}

Interpretation. Translational Maxwell distribution.

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Example 16. Isothermal expansion entropy

Definitions & inputs. Onemole,V2/V1=2.

  1. Choose the governing model and isolate the requested quantity.

    ΔS=Rln⁡2\Delta S=R\ln2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    8.314462618ln⁡28.314462618\ln2

  3. Evaluate the expression; the result uses the units shown.

    Result=5.763146 J/(mol K)\mathrm{Result}=5.763146\ \mathrm{J/(mol\,K)}

Interpretation. Ideal reversible end states.

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Example 17. Isothermal work

Definitions & inputs. Onemole300K,volume ratio2.

  1. Choose the governing model and isolate the requested quantity.

    W=RTln⁡2W=RT\ln2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    8.314462618(300)ln⁡28.314462618(300)\ln2

  3. Evaluate the expression; the result uses the units shown.

    Result=1728.944 J/mol\mathrm{Result}=1728.944\ \mathrm{J/mol}

Interpretation. Work done by gas in reversible expansion.

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Example 18. Relative energy fluctuation

Definitions & inputs. N10⁶monatomic particles.

  1. Choose the governing model and isolate the requested quantity.

    σE/U=2/(3N)\sigma_E/U=\sqrt{2/(3N)}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2/(3×106)\sqrt{2/(3\times10^6)}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0008164966 \mathrm{Result}=0.0008164966\ {}

Interpretation. Canonical ideal-gas result.

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Example 19. Metropolis uphill acceptance

Definitions & inputs. ΔE=2kBT.

  1. Choose the governing model and isolate the requested quantity.

    A=min⁡(1,e−βΔE)A=\min(1,e^{-\beta\Delta E})

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    e−2e^{-2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1353353 \mathrm{Result}=0.1353353\ {}

Interpretation. Symmetric proposal;auto correlation still matters.

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Example 20. Single Ising pair flip cost

Definitions & inputs. J1energy unit,no external field,aligned→antialigned.

  1. Choose the governing model and isolate the requested quantity.

    ΔE=J−(−J)\Delta E=J-(-J)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    22

  3. Evaluate the expression; the result uses the units shown.

    Result=2 \mathrm{Result}=2\ {}

Interpretation. One bond only,not the whole lattice.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.