Gibbs taqsimoti
Gibbs taqsimoti - bu termostat (muhit) bilan issiqlik muvozanatidagi zarrachalarning makroskopik termodinamik tizimining holatini taqsimlash . Klassik holatda, tarqatish zichligi teng bo'ladi: w ( X , a ) = 1 Z e − β H ( X , a ) , {\displaystyle w(X,a)={\frac {1}{Z}}e^{-\beta H(X,a)},} Bunda X {\displaystyle X} - o'rnatish 6 N {\displaystyle 6N} kanonik o'zgaruvchilar N {\displaystyle N} zarralar ( 3 N {\displaystyle 3N} koordinatalar va 3 N {\displaystyle 3N} impulslar), a {\displaystyle a} - tashqi parametrlar to'plami; H ( X , a ) {\displaystyle H(X,a)} tizimning Gamiltonianidir, β {\displaystyle \beta } -taqsimot parametridir. Qiymati Θ = 1 β {\displaystyle \Theta ={\frac {1}{\beta }}} tarqatish moduli deb ataladi. Tarqatish moduli ekanligini ko'rsatish mumkin Θ = k T {\displaystyle \Theta =kT} , bu yerda T {\displaystyle T} -mutlaq harorat, k {\displaystyle k} -Boltsman doimiysi. Z {\displaystyle Z} -normallashtirish sharti asosida aniqlangan parametrdir ∫ ( X ) w ( X , a ) d X = 1 {\displaystyle \int _{(X)}w(X,a)dX=1} , bundan kelib chiqadi Z = ∫ ( X ) e − β H ( X , a ) d X . {\displaystyle Z=\int _{(X)}e^{-\beta H(X,a)}dX.} Z {\displaystyle Z} -holat integrali deb ataladi. Gibbs taqsimotining quyidagi parametrlari ko'pincha qo'llaniladi: w ( X , a ) = e Ψ ( Θ , a ) − H ( X , a ) Θ , {\displaystyle w(X,a)=e^{\frac {\Psi (\Theta ,a)-H(X,a)}{\Theta }},} bu yerda Ψ ( Θ , a ) = − Θ ln ⁡ Z ( Θ , a ) {\displaystyle \Psi (\Theta ,a)=-\Theta \ln Z(\Theta ,a)} tizimning erkin energiyasi deb ataladigan narsadir. Kvant holatida energiya darajalarining hisoblangan to'plami qabul qilinadi va taqsimot zichligi o'rniga tizimning u yoki bu holatda bo'lish ehtimoli hisobga olinadi: W i = e Ψ − E i Θ . {\displaystyle W_{i}=e^{\frac {\Psi -E_{i}}{\Theta }}.} Normalizatsiya sharti shaklga ega ∑ i = 0 ∞ W i = 1 {\displaystyle \sum _{i=0}^{\infty }W_{i}=1} , shuning uchun Z = ∑ i = 0 ∞ e − E i Θ , {\displaystyle Z=\sum _{i=0}^{\infty }e^{-{\frac {E_{i}}{\Theta }}},} holatlarning integraliga o'xshash va holatlar yig'indisi yoki bo'linish funktsiyasi deb ataladi.