N个粒子的体系 ○，J,1。U Phase Transformation and Applications Page 16/66 U,V,N孤立体系/微正则系综的一种分布态 能级： e0N81、82、.、81、.、8r 粒子数： n0、n1n2、.、ni.、n- ∑n=N ∑n,6,=U i=0 i=0 这种分布态的微观状态数目o i=0 所有分布态微观状态数的总和称为体系的微观状态数Ω N! 2= i=0 i=0 Πn,! i=0 SJTU Thermodynamics of Materials Spring 2008 ©X.J.Jin Lecture 19 Statistical
Phase Transformation and Applications S. J. T. U. SJTU Thermodynamics of Materials Spring 2008 © X. J. Jin Lecture 19 Statistical Page 16/66 N个粒子的体系 U，V，N孤立体系/微正则系综 的 一种分布态 Nn r i ∑ i = =0 Un r i ∑ ii = =0 ε 能级： ε0、ε1、ε2、…、εi、…、 εr 粒子数： n0、n1、n2、…、ni、…、 nr 这种分布态的微观状态数目ω ∏ = r i i n N 0 ! ! 所有分布态微观状态数的总和称为体系的微观状态数Ω ∑ ∏ ∑ = = = ==Ω m i r i i m i i n N 0 0 0 ! ! ω
微观态存在的机会和对Ω的贡献 S.J.T.U. Phase Transformation and Applications Page 17/66 平衡态？稳态则常伴有能量流 一个封闭系统经过足够长的时间自动达到所有宏观状 态量都不再变化的状态。 基本假定 ·每个微观态等几率出现(U,V,N确定的微正则系综) 最可几分布所代表的状态就是体系的平衡态(U、V和 N确定) SJTU Thermodynamics of Materials Spring 2008 ©X.J.Jin Lecture 19 Statistical
Phase Transformation and Applications S. J. T. U. SJTU Thermodynamics of Materials Spring 2008 © X. J. Jin Lecture 19 Statistical Page 17/66 微观态存在的机会和对Ω的贡献 • 平衡态？/稳态则常伴有能量流 一个封闭系统经过足够长的时间自动达到所有宏观状 态量都不再变化的状态。 • 基本假定 • 每个微观态等几率出现（U,V,N确定的微正则系综） • 最可几分布所代表的状态就是体系的平衡态（U、V和 N确定）
Boltzman假定 S.J.T.U. Phase Transformation and Applications Page 18/66 ·熵和微观状态数之间的联系 混乱程度：熵、微观状态数 S-kIn k为Boltzmann常数 微正则 正则 SJTU Thermodynamics of Materials Spring 2008( X.J.Jin Lecture 19 Statistical
Phase Transformation and Applications S. J. T. U. SJTU Thermodynamics of Materials Spring 2008 © X. J. Jin Lecture 19 Statistical Page 18/66 Boltzman假定 • 熵和微观状态数之间的联系 混乱程度：熵、微观状态数 = kS lnΩ k为Boltzmann常数 微正则 正则
Isolated System and Boltzmann Hypothesis S.J.T.U. Phase Transformation and Applications Page 19/66 Microcanonical ensemble An isolated system with N particals in a volume,V,with a fixed energy,E Premise:all microstates are equally probable Boltzmann hypothesis The entropy of a system is linearly related to the logarithm of S=kIn thermodynamic probability the number of different ways that macro-configuration can be achieved. N! Q= Entropy of mixing of two components N!Nu! SJTU Thermodynamics of Materials Spring 2008( X.J.Jin Lecture 19 Statistical
Phase Transformation and Applications S. J. T. U. SJTU Thermodynamics of Materials Spring 2008 © X. J. Jin Lecture 19 Statistical Page 19/66 Isolated System and Boltzmann Hypothesis Microcanonical ensemble An isolated system with N particals in a volume, V, with a fixed energy, E Premise: all microstates are equally probable Boltzmann hypothesis The entropy of a system is linearly related to the logarithm of Ω Ω：thermodynamic probability the number of different ways that macro-configuration can be achieved. !! ! ΠΙ =Ω NN N Entropy of mixing of two components = kS lnΩ
统计热力学基础：经典与统计 S.J.T.U. Phase Transformation and Applications Page 20/66 统计热力学 df(@AB) d@A.B 熵的统计表达式 f(AB)A.B f(O)=f(0a)+f(0B) Inf(@1a)=In-+Ink 04,B=0A0B f(@)=f(@@g)=f(@)+f(@g) f'(04,B)= DA.B DBf'(04·0B)=f'(0A) f(@AB)=kIn@48+C 040Bf"(040B)+f'(040B)=0 S=kIn@48+C 04,Bf"(0A.B)+f'(0LB)=0 S=klno SJTU Thermodynamics of Materials Spring 2008 ©X.J.Jin Lecture 19 Statistical
Phase Transformation and Applications S. J. T. U. SJTU Thermodynamics of Materials Spring 2008 © X. J. Jin Lecture 19 Statistical Page 20/66 统计热力学基础：经典与统计 统计热力学 熵的统计表达式 )()()( A B ω = ω + fff ω ω ω ωBABA = ⋅ , )()()()( BA A B ω = ff ω ⋅ω = ω + ff ω )(')(' BAB A ω f ω ⋅ω = f ω ⋅ + ⋅ = 0)(')('' BA BA BA ω ω f ω ω f ω ω 0)(')('' ω ,BA ω ,BA + ff ω ,BA = BA BA BA BA d f df , , , , )(' )(' ω ω ω ω = f k BA BA ln 1 ln)('ln , , += ω ω BA BA k f , , )(' ω ω = Ckf ω ,BA = ln)( ω ,BA + = lnω ,BA +CkS = kS lnω