ABSTRACT
From a theoretical point of view, the simplest system is an isolated system since all interactions with the external environment are neglected. The energy https://www.w3.org/1998/Math/MathML"> E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the number of particles https://www.w3.org/1998/Math/MathML"> N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and the volume https://www.w3.org/1998/Math/MathML"> V https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of the system are then fixed. With these chosen external parameters https://www.w3.org/1998/Math/MathML"> ( N , V , E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the statistical ensemble, called microcanonical ensemble, is completely defined. The probability distribution function of the associated microstates is then the result of a postulate, known as the “fundamental” postulate, which completes the first postulate of equality of the temporal and ensemble averages (Equation (1.28)). The role of the microcanonical ensemble is central in statistical physics. Firstly, because the energy and the number of particles are quantities defined at all scales, which therefore allow the control of the transition from microscopic to macroscopic in the limit https://www.w3.org/1998/Math/MathML"> N ≫ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Secondly, because the other statistical ensembles, characterised by different external parameters, derive from the microcanonical ensemble.
After stating the fundamental postulate (Section 2.1), the enumeration process of the accessible microstates, a central calculation in the microcanonical ensemble allowing the deduction of the system’s thermodynamic properties, will be explained (Section 2.1). Next, it will be shown that a macroscopic internal variable is in general distributed according to an extremely sharp Gaussian distribution. In other words, fluctuations around the average value are negligible (Section 2.3). Finally, the foundations of statistical physics and the justification of the two postulates will be discussed (Section 2.4).
