GP

Statistical Mechanics

Statistical mechanics is the extrapolation of quantitative properties from systems of large numbers of particles (on the order of \(~10^{23}\) particles). The general approach is to specify some system in terms of it's Hamiltonian, and then to apply some corresponding treatment of the system as a statistical ensemble defined by a specific set of macroscopic properties. Thus, statistical mechanics allows us to model macroscopic systems with large numbers of constituents quantitatively, by averaging over and modeling systems in terms of macroscopic properties. Thermodynamics is a subset of this more general set of tools.

Contents

Thermodynamic Relations

Perhaps the most intuitive thermodynamic function is the internal energy of a system. For a general system, the internal energy is defined as a function of the entropy, volume, and number of particles. Taking the exterior derivative of this then gives the relation

\begin{align} dU(S,V,N)&=\frac{\partial U}{\partial S}&dS & +&\frac{\partial U}{\partial V}&dV&+&\frac{\partial U}{\partial N}&dN&\\ &=T&dS&+&-P&dV&+&\mu &dN& \ \ \tiny{\text{(Euler Relation)}} \end{align}
Immediately, we see that the properties of temperature \(T\), pressure \(P\), and chemical potential \(\mu\) may be defined in terms of derivatives of the internal energy of the system. Such properties that are derivatives of the internal energy are termed intensive properties, as they are intrinsic to any arbitrarily selected portion of the overall system (i.e., any subsystem). However, the variables that determine the energy are all neccesarily extensive, in that volume \(V\), entropy \(S\), and the number of particles \(N\) are a measure of the total system, and are not properties inherent in any particular entity of those comprising the system. Particularly, extensive properties all meet the scalability condition of \(U\) in that \( \lambda U(S,V,N)= U(\lambda S,\lambda V,\lambda N)\).

Intrinsic Properties

As discussed above, intensive properties are those that characterize not the total system but inherent properties of any (adequately sized) subsystem contained within. As such, they're generally derivatives of state functions with respect to extensive properties of the system.

\begin{align} \frac{\partial U}{\partial S}&=&T\\ \frac{\partial U}{\partial V}&=&-P\\ \frac{\partial U}{\partial N}&=&\mu\\ \end{align}

Extrinsic Properties

Extrinsic properties are those that only describe the collected system in it's entirety. That is, they are not properties inherent in any subsystem contained within. Rather, they're physical properties that only describe the system's overall extent and content. These properties generally include \(U\) (internal energy), \(S\) (entropy), \(V\) (volume), and \(N\) (the number of particles); as well as other definable properties of this nature.

Other Common Properties

Some other common properties that may be defined for thermodynamic systems are expansion coefficients, compressibilities, and specific heats. Generally, they're specified with respect to some other state variable that doesn't appear explicitly in the defining relation itself. This can be taken as a signal that this component is being held constant, but it's more to specify that, in the relevant context, we're treating the relevant state function as a function of the variables referenced (i.e. the one being differentiated w.r.t. and the one specified externally). Defining relations for some of these properties are specified below for convenience.

Specific Heats

Specific heats describe the differential relationship between entropy and internal energy of the system.

\begin{align} c_v&=T\left(\frac{\partial S}{\partial T}\right)_V=\left(\frac{\partial S}{\partial U}\right)_V\\ c_p&=T\left(\frac{\partial S}{\partial T}\right)_P=\left(\frac{\partial S}{\partial U}\right)_P \end{align}

Compressibilities

Compressibilities describe how the volume of a system changes under changes in pressure.

\begin{align} \kappa_T&=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T \end{align}

Expansion Coefficients

Coefficients of volume expansion describe how volume changes under changes in temperature.

\begin{align} \alpha&=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P,N} \end{align}

Laws of Thermodynamics

First Law

The first law of thermodynamics is a statement of energy conservation. It essentially states that the internal energy of a system is only depepndent on the work done on the system and the heat absorbed by the system.

$$ \Delta U = \Delta Q - \Delta W $$

Second Law

The second law defines the direction of time, in that it claims the sum of entropies of isolated systems is less than or equal to the entropy of the total system given the systems are then allowed to interact over time.

$$ \delta Q = TdS $$

Third Law

The third law specifies that the entropy of any system approaches a constant value at zero temperature. Nernst's postulate further states that this constant value is exactly zero, but this has been proven empirically false in certain systems, such as super cooled glasses.

$$ \lim_{T\rightarrow 0}=C $$

Ensembles

A statistical ensemble, or just ensemble, is a particular approach for applying the methods of statistical mechanics. Effectively, ensembles are distributions in phase space, defined by a set of variables in a particular system completely specified by it's Hamiltonian.

To to that end, ensembles are predetermined strategies for applying the methods of thermodynamics, and statistical mechanics at large, to systems characterized by a certain set of extensive or intensive properties. They can be thought of as the way in which we may distribute a certain set of properties across all the potential arrangements in phase space that satisfy such properties.


Micro-canonical Ensemble \( (E,N,V)\)

The micro-canonical ensemble can be used to model isolated or completely contained systems. That is, systems in which their total energy, number of particles, and volume can, at any instant, be completely specified.

General Approach in the Micro-Canonical Ensemble

Classical

Suppose you are given some macroscopic system, specified by it's Hamiltonian, and know the system's energy. From this relation for energy and the expectation that it is well defined for all potential arrangements of the system, we may calculate from it the number of accessible states \(N_{\Gamma_a}\) in the relevant phase space \(\Gamma\) for any total internal energy \(E\).

$$ N_{\Gamma_a}(E) = \int \frac{d\Gamma}{\Gamma_0} \delta (H(p_i,q_i)-E) $$
where the number of accesible states is really the volume of accessible phase space partitioned into quantized packets arbitrarily (though motivated by quantum mechanics). From this number of accessible states, we may derive the entropy for the system via the relation below and thence calculate arbitrary thermodynamic quantities.
$$ S(U,V,N)=k_B lnN_{\Gamma_a} $$

Thermodynamic Relations Relevant to Micro-Canonical Ensemble

Since the entropy of the system is the only result of this treatment, we may only perform thermodynamics in the micro-canonical ensemble. Nevertheless, it may be powerful for isolated systems. Relevant relations related to entropy \(S\) are specified below for convenience.

\begin{align} \frac{\partial S}{\partial U}&=&\frac{1}{T}\\ \frac{\partial S}{\partial V}&=&-\frac{P}{T}\\ \frac{\partial S}{\partial N}&=&\frac{\mu}{T}\\ \end{align}

Quantum Micro-Canonical Ensemble

In the quantum mechanical formalism, our micro-canonical ensemble is described as a density matrix of the form given below:

$$ \rho=\sum_n p_n \vert\psi_n\rangle\langle\psi_n\vert $$
with \(p_n=1/D_n\) where \(D_n\) is the degeneracy of the nth energy level.


Canonical Ensemble \( (T,V,N)\)

The canonical ensemble is an approach suited for application to systems distinguishable from their surroundings, but which interact with their surrounding 'environment'. That is, this treatment may be applied to systems with a definite number of particles, volume, and temperature, but may still exchange energy within a larger system. Thus, the energy of our subsystem under analysis isn't entirely well-defined at an arbitrary time and it is neccessary to specify, or depend on, an intensive property of the system when defining it, hence the relevance of temperature.

General Approach in the Canonical Ensemble

Classical

The general approach to modeling systems in the canonical ensemble is to solve for the particular partition function \(Z\) of the system, from which we may perform statistical mechanics. If one wishes to model such a system thermodynamically, the general approach then is to derive the Helmholtz free energy from \(Z\) and derive other thermodynamical properties from there.

The partition function is defined for a system's Hamiltonian that may depend on the (specified) volume and number of particles of the system; the partition function is then only a function of temperature. Explicity, the partition function in the canonical ensemble is given by
$$ Z(\beta) = \int \frac{d\Gamma}{\Gamma_0} e^{-\beta H} $$
and then from here, the Helmholtz free energy of the system is easily found via
$$ F(T,V,N) = -k_B T ln Z = U - TS $$

Statistical Relations Relevant to Canonical Ensemble

From the partition function, we may define the average value of energy (denoted by a bar above the operator) via

\begin{align} \overline{E}&=\frac{1}{Z}\int d\Gamma H\left( {p_i,q_i}\right)\ e^{-\beta H}\\ &=-\frac{\partial ln Z}{\partial \beta}\\ &=-\frac{\partial (\beta F)}{\partial \beta}=U\\ \end{align}
Then, in the general case for cumulants:
\begin{align} \overline{\Delta E^n}&=(-1)^n\frac{\partial ^n ln Z}{\partial \beta^n}\\ &=(-1)^n\frac{\partial ^n(\beta F)}{\partial \beta^n}\\ &=(-1)^{n-1}\left(\frac{\partial }{\partial \beta}\right)^{n-1}\overline{E} \end{align}

Thermodynamic Relations Relevant to Canonical Ensemble

Since the Helmholtz free energy is the system of state relevant to treatments in the canonical enesemble, relevant relations are specified below.

\begin{align} \frac{\partial F}{\partial T}&=&-S\\ \frac{\partial F}{\partial V}&=&-P\\ \frac{\partial F}{\partial N}&=&\mu\\ \end{align}
Another important relation is that of specific heat, which is given below for the canonical ensemble.
$$ c_v=\frac{\partial \overline{E}}{\partial T} $$

Quantum Canonical Ensemble

In the quantum mechanical formalism, our canonical ensemble is described as a density matrix of the form given as:

$$ \rho=\frac{1}{Z}\sum_n e^{-\beta E_n} \vert\psi_n\rangle\langle\psi_n\vert $$
with the relevant partition function \(Z\) defined as
$$ Z=Tr(e^{-\beta H}) $$


Grand Canonical Ensemble \( (T ,V,\mu )\)

An approach to statistical mechanics apt for subsystems which further exchange particles with their surrounding environment, but are chemically indistinguishable from those elements of it's environment. Essentially, we replace dependence on the number of particles \(N\) with dependence on inherent chemical properties of these particles, namely their chemical potential \(\mu\).

General Approach in the Grand-Canonical Ensemble

\begin{align} \mathcal{Z}(\beta) & = \sum_{N_i=0}^{\infty}e^{\beta\mu N_i}\int \frac{d\Gamma}{\Gamma_0} e^{-\beta H}\\ \ \ \\ & = \sum_{N_i=0}^{\infty}e^{\beta\mu N_i} Z_N(\beta) \end{align}

The general approach in the grand canonical ensemble is to solve for the particular partition function \(\mathcal{Z}\) of the system given by the relation above. From here the grand potential of the system is easily found via

$$ \Omega(T,V,\mu ) = -k_B T ln\mathcal{Z} = U-TS-\mu N $$

Statistical Relations Relevant to Grand-Canonical Ensemble

From the grand-canonical partition function, we may define the average value of an arbitrary operator via

$$ \overline{A}=\frac{1}{\mathcal{Z}}\sum_{N=0}^{\infty}\int d\Gamma e^{-\beta(H_N-\mu N)}A_N $$

Thermodynamic Relations Relevant to Grand-Canonical Ensemble

The grand potential \(\Omega \) is the system of state relevant to the grand-canonical enesemble, and as such, relevant relations are specified below.

\begin{align} \frac{\partial \Omega}{\partial T}&=&-S\\ \frac{\partial \Omega}{\partial V}&=&-P\\ \frac{\partial \Omega}{\partial \mu}&=&-N\\ \end{align}

Distributions

Several probability distributions are particularly relevant for statistical mechanics. These generally include: the Maxwell-Boltzmann Distribution, relevant for classical thermal applications; the Fermi-Dirac Distribution, applicable to Fermionic quantum systems; and the Bose-Einstein Distribution, related to Bosonic quantum systems.

Maxwell-Boltzmann

The Maxwell-Boltzmann distribution describes the thermal distribution of the velocity of classical gasses. Note that it only applies in thermal equilibirium and to gases that aren't exhibiting some macroscopic flow.

\begin{align} f(v)=\left(\frac{m}{2\pi k_BT}\right)^{3/2}e^{-\frac{mv^2}{2k_BT}} \end{align}

Fermi-Dirac

Fermi-Dirac distributions are used to describe the probability of energy states being occupied for thermal Fermionic systems. Fermions obey the Pauli exclusion principle, as they have non-integer spin, which immediately resolves all degeneracies.

\begin{align} f(\epsilon _i)=\frac{1}{e^{(\epsilon_i-\mu)/k_BT}+1} \end{align}

Sommerfeld Expansion

In the context of Fermi-Dirac statistics, we often come across integrals of the form:

\begin{align} \int_{-\infty}^{\infty}f(\epsilon)\phi(\epsilon)d\epsilon. \end{align}
These are, generally, hard to solve. However, a convenient asymptotic expansion of the above is available for small temperatures and is known as the Sommerfeld expansion. The first few terms of the Sommerfeld expansion are given as:
\begin{align} \int_{-\infty}^{\infty}f(\epsilon)\phi(\epsilon)d\epsilon\approx^{\text{low } T}\int_{-\infty}^{\mu}\phi(\epsilon)d\epsilon +\frac{\pi^2}{6}\left(\frac{1}{\beta}\right)^2\frac{d\phi}{d\epsilon}\big\vert_{\mu}+... \end{align}

Bose-Einstein

Bose-Einstein distributions are used to describe the occupation of states of thermal Bosonic systems, which, in contrast to Fermions, can have several quanta occupying the same state.

\begin{align} f(\epsilon _i)=\frac{g_i}{e^{(\epsilon_i-\mu)/k_BT}-1} \end{align}
Here, it is important that the chemical potential \(\mu\) is always less than the ground state energy of the system.

Moments & Cumulants of PDFs

Moments

Moments may be calculated via

$$ M_n=\int dx\ x^nP(x). $$
The moment generating function may be derived as
\begin{align} M(t)&=\int dx\ e^{xt}P(x)\\ &=\int dx\ \sum_{n=0}^{\infty}\frac{1}{n!}x^nt^nP(x)\\ &=\sum_{n=0}^{\infty}M_n\frac{t^n}{n!} \end{align}

Cumulants

The cumulant generating function may be evaluated as

$$ K(t)=\ln M(t). $$

Other Common Relations

Gaussian integral:

$$ \int_{-\infty}^{\infty}dx\ e^{-a(x-b)^2}=\sqrt{\frac{\pi}{a}} $$

Stirling's Approximation (Large N):

$$ \ln N!=N\ln N -N $$

Geometric Series:

$$ \sum_{k=0}^{\infty}r^k=\frac{1}{1-r} $$ $$ \sum_{k=0}^{n}r^k=\frac{1-r^{n+1}}{1-r} $$

Binomial Expansion:

$$ (x+1)^{\nu}=\sum_{k=0}^{\infty}\begin{pmatrix}\nu \\ k\end{pmatrix}x^k $$ $$ (x+a)^{\nu}=\sum_{k=0}^{\infty}\begin{pmatrix}\nu \\ k\end{pmatrix}x^ka^{\nu-k} $$

Volume of n-ball with radius R:

$$ V_n(R)=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}R^n $$
Surface area of n-sphere with radius 1:
$$ S_{n}=2\pi V_{n-1} $$