Formulaire par cours

\(\displaystyle\bar f = \sum_{i=1}^{n} p_i f_i\)

\(\overline{f+g} = \bar f + \bar g\) ; \(\overline{(cf)} = c\bar f\) pour \(c\in\mathbb R\)

\(\overline{fg} = \bar f\cdot\bar g\)

\(\bar n = pN\)

\(p + q = 1\)

Critères indépendants : \(P = P_1\,P_2\)

\(\displaystyle C_N^n = \binom{N}{n} = \frac{N!}{n!\,(N-n)!}\)

\(\displaystyle A_N^n = \frac{N!}{(N-n)!} = N(N-1)\cdots(N-n+1)\)

\(N!\) permutations de \(N\) éléments distincts

\(\displaystyle m! \approx \sqrt{2\pi m}\left(\frac{m}{e}\right)^{m}\)

\(\ln(m!) \approx m\ln m - m + \tfrac12\ln(2\pi m)\)

\(m! \approx 10^{\,0{,}4343\,(m\ln m - m + \frac12\ln 2\pi m)}\)

\(\displaystyle\sigma = \sqrt{\overline{(f-\bar f)^2}}\) ; \(\text{Variance} = \sigma^2 = \overline{f^2}-\bar f^{\,2}\)

\(\text{Variance}(f+g) = \text{Variance}(f)+\text{Variance}(g)\)

\(\sigma = \sqrt{Npq}\) ; \(\displaystyle\frac{\sigma}{\bar n} = \sqrt{\frac{q}{pN}}\propto\frac{1}{\sqrt N}\)

\(\displaystyle P_N(n) = \frac{1}{\sqrt{2\pi}\,\sigma}\,e^{-\frac{(n-\bar n)^2}{2\sigma^2}} = A\,e^{-B(n-\bar n)^2}\)

\(\displaystyle A = \frac{1}{\sqrt{2\pi}\sigma} = P_N(\bar n)\),   \(\displaystyle B = \frac{1}{2\sigma^2}\)

\(p_\sigma\approx0{,}68\), \(p_{2\sigma}\approx0{,}95\), \(p_{3\sigma}\approx0{,}997\)

\(\overline{S_n} = n\,\bar s\)  et  \(\sigma_n = \sigma\sqrt n\)

\(\displaystyle p_n(L) = C_n^{\,k}\,p_d^{\,k}p_g^{\,n-k}\), \(k = \frac{L+n}{2}\)

Pas 3D de longueur \(a\), direction aléatoire : \(\bar s_z = 0\), \(\sigma_z = a/\sqrt3\)

\(u(x) = u_0 + \tfrac12\kappa x^2\)

\(\displaystyle\varepsilon_c = \frac{p^2}{2m} = \frac{p_x^2+p_y^2+p_z^2}{2m}\) ; \(\displaystyle\varepsilon_{r,i} = \frac{L_i^2}{2I_i}\)

\(\displaystyle\varepsilon = u_0 + \sum_{\text{ddl}}\beta\xi^2\),   \(\displaystyle\varepsilon_{therm} = \sum_{\text{ddl}}\beta\xi^2\)

Gaz monoatomique : \(\nu = 3\) (translations)

Gaz diatomique : \(\nu = 5\) (3 translations + 2 rotations)

Solide : \(\nu = 6\) (3 cinétiques + 3 potentiels)

\(\overline{\varepsilon_{\text{ddl}}} = \tfrac12kT\) ; \(\displaystyle\varepsilon_{therm} = \frac{\nu}{2}kT\)

\(\displaystyle E_{therm} = \frac{N\nu}{2}kT\) ; \(\displaystyle E = Nu_0 + \frac{N\nu}{2}kT\)

\(\displaystyle \frac{\overline{p^2}}{2m} = \frac32kT\ \Rightarrow\ p_{rms} = \sqrt{3mkT},\ v_{rms} = \sqrt{\frac{3kT}{m}}\)

\(\Delta W = F\Delta x\) ; \(\Delta W = p\Delta V\)  (\(\displaystyle\Delta W = \int p\,dV\))

\(\Delta W = -\vec B\cdot\Delta\vec\mu\) ; \(\Delta W = -\vec E\cdot\Delta\vec p\)

\(\Delta E = \Delta Q - \Delta W + \mu\Delta N\)

\(\Delta E_1 = \mu_1\Delta N\), \(\Delta E_2 = -\mu_2\Delta N\)

\(\Omega = \omega\times\omega\times\cdots = \omega^N\)

\(\displaystyle\Omega = \frac{\omega^N}{N!}\approx\left(\frac{e\omega}{N}\right)^N = \omega_c^{\,N}\),   \(\displaystyle\omega_c = \frac{e\omega}{N}\)

\(\Omega(E,\Delta E) = g(E)\,\Delta E\) ; \(\Omega_{A+B} = \Omega_A\Omega_B\)

\(\Delta x\,\Delta p_x\ge h\)

\(\displaystyle\omega = \frac{[x][p_x]}{h}\) (1D) ; \(\displaystyle\omega = \frac{V_rV_p}{h^3}\) (3D)

\(\displaystyle\omega = \frac{V}{h^3}\iiint dp^3 = \frac{4\pi V}{3h^3}\left(p_2^3 - p_1^3\right)\)

Gaz : \(\displaystyle\omega_c\propto\frac{V}{N}\left(\frac{E_{therm}}{N\nu}\right)^{\nu/2}\),   \(\displaystyle\Omega = \omega_c^N\propto\left(\frac{V}{N}\right)^N\left(\frac{E_{therm}}{N\nu}\right)^{N\nu/2}\)

Solide : \(\displaystyle\omega_c\propto\left(\frac{E_{therm}}{N\nu}\right)^{\nu/2}\),   \(\displaystyle\Omega = \omega_c^N\propto\left(\frac{E_{therm}}{N\nu}\right)^{N\nu/2}\)

\(g(E)\propto\) mêmes expressions ; avec \(E_{therm} = N\nu kT/2\) : \(g\propto(V/N)^N(kT/2)^{N\nu/2}\) (gaz)

\(E_1 + E_2 = E_0\) ; \(\Omega_0 = \Omega_1\Omega_2 \propto E_1^{n_1/2}(E_0 - E_1)^{n_2/2}\)

\(\displaystyle E_1^* = \frac{n_1}{n_1+n_2}E_0\)

\(S = k\ln\Omega\) ; \(S_0 = S_1 + S_2\)

\(\displaystyle\left(\frac{\partial S}{\partial V}\right)_{E,N} = \frac{p}{T}\) ; \(\displaystyle\left(\frac{\partial S}{\partial N}\right)_{E,V} = -\frac{\mu}{T}\) ; \(dQ = T\,dS\)

\(\displaystyle\Delta S = S_f - S_i = k\ln\Omega_f - k\ln\Omega_i = k\ln\frac{\Omega_f}{\Omega_i}\)

\(\displaystyle\frac{\Omega_f}{\Omega_i} = e^{\Delta S/k} = e^{\Delta Q/kT}\)

\(\displaystyle C_y = \left(\frac{dQ}{dT}\right)_y\) ; molaire \(C = C_y/n\) ; chaleur spécifique \(c = C_y/m\)

\(\displaystyle\Delta S = \int\frac{C\,dT}{T} = m\,c\ln\frac{T_f}{T_i}\) ; \(\displaystyle\Delta S_{phase} = \frac{Q_{latente}}{T_{phase}}\)

\(S(T = 0\ \text{K}) = 0\) J/K

\(\displaystyle S(T) = \int_0^T\frac{dQ}{T'} = \int_0^T\frac{C_p(T')\,dT'}{T'}\)

\(\displaystyle\left(\frac{\partial S}{\partial T}\right)_p = \frac{C_p}{T}\) ; \(\displaystyle\left(\frac{\partial S}{\partial V}\right)_T = \frac{\beta}{\kappa}\) ; \(\displaystyle\left(\frac{\partial T}{\partial V}\right)_S = -\frac{T\beta}{C_V\kappa}\)

\(\displaystyle\left(\frac{\partial S}{\partial T}\right)_V = \frac{C_V}{T}\) ; \(\displaystyle\left(\frac{\partial S}{\partial V}\right)_p = \frac{C_p}{TV\beta}\) ; \(\displaystyle\left(\frac{\partial T}{\partial V}\right)_p = \frac{1}{V\beta}\)

\(\displaystyle\left(\frac{\partial S}{\partial p}\right)_T = -V\beta\) ; \(\displaystyle\left(\frac{\partial T}{\partial p}\right)_S = \frac{TV\beta}{C_p}\) ; \(\displaystyle\left(\frac{\partial p}{\partial V}\right)_S = -\frac{C_p}{VC_V\kappa}\)

\(\displaystyle\left(\frac{\partial S}{\partial p}\right)_V = \frac{C_V\kappa}{T\beta}\) ; \(\displaystyle\left(\frac{\partial T}{\partial p}\right)_V = \frac{\kappa}{\beta}\) ; \(\displaystyle\left(\frac{\partial p}{\partial V}\right)_T = -\frac{1}{V\kappa}\)

\(dE = (C_p - pV\beta)\,dT\)

gaz parfait : \(W = p\Delta V = nR\Delta T\) ; \(Q = nC_p\Delta T\) ; \(\Delta E = nC_V\Delta T\) ; \(\displaystyle\Delta S = nC_p\ln\frac{T_f}{T_i}\) ; \(\displaystyle\frac{T_f}{T_i} = \frac{V_f}{V_i}\)

\(\displaystyle dE = \left(\frac{T\beta}{\kappa} - p\right)dV\) ; \(\displaystyle dE = \left(T - \frac{p\kappa}{\beta}\right)dS\)

gaz parfait : \(\Delta E = 0\) ; \(\displaystyle Q = W = nRT\ln\frac{V_f}{V_i}\) ; \(\displaystyle\Delta S = nR\ln\frac{V_f}{V_i}\) ; \(\displaystyle\frac{V_f}{V_i} = \frac{p_i}{p_f}\)

\(\displaystyle\frac{dT}{T} = \frac{V\beta}{C_p}dp\) ; \(\displaystyle\frac{dT}{T} = -\frac{\beta}{\kappa C_V}dV\) ; \(\displaystyle\frac{dV}{V} = -\frac{\kappa C_V}{C_p}dp\)

\(pV^\gamma = \) cste où \(\displaystyle\gamma = \frac{\nu+2}{\nu}\) ; \(TV^{\gamma-1} = \) cste ; \(T\,p^{1/\gamma-1} = \) cste

\(Q = 0\) ; \(\Delta S = 0\) ; \(\displaystyle W = -\Delta E = \frac{p_iV_i - p_fV_f}{\gamma-1} = nC_V(T_i - T_f)\)

\(\displaystyle\frac{\Delta Q}{\Delta t} = -kA\frac{\Delta T}{\Delta x}\) ; \(\displaystyle j_Q = -k\frac{\Delta T}{\Delta x}\)

Résistance thermique \(\displaystyle R = \frac{\Delta x}{kA}\) ; \(\displaystyle\dot Q = \frac{\Delta T}{R}\) ; série \(R = \sum R_i\) ; parallèle \(\displaystyle\frac1R = \sum\frac1{R_i}\)

\(\displaystyle\frac{\partial E}{\partial t} = -A\frac{\partial j_Q}{\partial x}\Delta x\) ou \(\displaystyle\frac{\partial\rho_E}{\partial t} = -\frac{\partial j_Q}{\partial x}\)

\(\displaystyle\frac{\partial^2T}{\partial x^2} = \frac1K\frac{\partial T}{\partial t}\) où \(\displaystyle K = \frac{k}{\rho_m c}\)

\(pV = nRT\), \(C_V = \frac{\nu}{2}R\), \(C_p = C_V + R\), \(\gamma = (\nu+2)/\nu\) ; adiabatique \(W = (p_iV_i - p_fV_f)/(\gamma-1)\)

\(\displaystyle e_{Carnot} = \frac{Q_h - Q_c}{Q_h} = 1 - \frac{Q_c}{Q_h} = 1 - \frac{T_c}{T_h}\)

Pour un cycle de Carnot \(\displaystyle\frac{Q_h}{T_h} = \frac{Q_c}{T_c}\)

Série : \(R_{total} = R_{conv,1} + R_{cond,1} + \cdots + R_{conv,2}\)

Parallèle : \(\displaystyle\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}\)

\(\displaystyle\dot Q = \frac{T_i - T_j}{R_{total,\,i-j}}\)

\(P_s = C e^{-\beta\varepsilon_s}\) avec \(\beta = 1/kT\) et \(\displaystyle C = \frac{1}{\sum_s e^{-\beta\varepsilon_s}}\)

\(\displaystyle \frac{P_i}{P_j} = e^{-\beta(\varepsilon_i - \varepsilon_j)}\)

Niveaux \(\varepsilon_n = n\varepsilon\) : \(C = 1 - e^{-\beta\varepsilon}\)

\(P_n = C e^{-n\beta(\varepsilon - \mu)} = Ce^{-nx}\), \(x = \beta(\varepsilon - \mu)\), \(\displaystyle C = \frac{1}{\sum_n e^{-nx}}\)

Fermions (\(n = 0, 1\)) : \(\displaystyle C = \frac{1}{1 + e^{-x}}\) ; bosons (\(n \ge 0\), \(x \gt 0\)) : \(C = 1 - e^{-x}\)

\(P(\varepsilon_s) = n_s P_s\) ; \(\displaystyle \frac{P(\varepsilon_i)}{P(\varepsilon_j)} = \frac{n_i}{n_j}e^{-\beta(\varepsilon_i - \varepsilon_j)}\)

\(\displaystyle P_{bande} = \int_{bande} g(\varepsilon)P(\varepsilon)\,d\varepsilon\)

\(\displaystyle \bar\varepsilon = \sum_s P_s\varepsilon_s = \frac{\int e^{-\beta b\xi^2} b\xi^2\,d\xi}{\int e^{-\beta b\xi^2}\,d\xi} = \frac{1}{2\beta} = \frac{1}{2}kT\)

\(\varepsilon = b|\xi|^{\alpha}\) : \(\bar\varepsilon = kT/\alpha\)

\(\displaystyle \bar n = \sum_n nP_n = -\frac{\partial}{\partial x}\ln\!\left(\sum_n e^{-nx}\right)\), \(x = \beta(\varepsilon - \mu)\)

\(\displaystyle \bar n_{fermions} = \frac{1}{e^{x} + 1}\) ; \(\displaystyle \bar n_{bosons} = \frac{1}{e^{x} - 1}\) ; \(\displaystyle \bar n = \frac{1}{e^{\beta(\varepsilon-\mu)} \pm 1}\)

\(\displaystyle \bar n_{classique} = NP_s = NCe^{-\beta\varepsilon_s}\) ; \(\displaystyle e^{\beta\mu} = \frac{N}{\sum_s e^{-\beta\varepsilon_s}}\)

\(\varepsilon - \mu \gg kT\) : \(\bar n_{cl} \approx \bar n_B \approx \bar n_F \approx e^{-\beta(\varepsilon - \mu)}\)

\(\varepsilon - \mu \lesssim kT\) : \(\bar n_{bosons} \gt \bar n_{classique} \gt \bar n_{fermions}\)

\(\displaystyle \frac{V_rV_p}{h^3} \gg N\) → \(\displaystyle \frac{rp}{h} \gg 1\) avec \(\displaystyle r \approx \left(\frac{V}{N}\right)^{1/3}\), \(p \approx \sqrt{3mkT}\)

\(\displaystyle N = \sum_s \bar n_s\) ; faible densité : \(\displaystyle \mu = kT\ln\frac{N}{\sum_s e^{-\beta\varepsilon_s}}\)

Bosons dégénérés : \(\mu \approx -kT/N\) ; fermions dégénérés : \(\displaystyle \mu_{T=0} = \varepsilon_F \approx \frac{h^2}{2m}\left(\frac{N}{V}\right)^{2/3}\)