LOG#186. Polylogarithmic condensates.

Today, I return to my best friends. The polylogs! Are you ready for polylog wars? The polylogarithm can be represented by the next integral:

(1)   \begin{equation*} Li_s(s)=\dfrac{z}{\Gamma (s)}\int_0^1\left(-\ln u\right)^{s-1}\dfrac{du}{1-uz} \end{equation*}

As you know, if you follow my blog, you have

    \[Li_1 (z)=\ln (1-z)\]

    \[Li_0 (z)=\dfrac{z}{1-z}\]

    \[z\dfrac{d}{dz}Li_{s+1} (z)=\dfrac{d}{\ln z}\left(Li_{s+1}(z)\right)=Li_s (z)\]

Off-topic: I have fallen in love again (with another mathematical function!). The Lambert W-function is the inverse of Xe^X=Y, so it can be inverted W(z)e^{W(z)}=z, and this is important for solving infinite tetrations ^\infty z=c via

(2)   \begin{equation*} \boxed{c=-\dfrac{W(-\ln z)}{\ln z}} \end{equation*}

Even the most simple polylogarith, the Riemann zeta function, serves for solving in closed form lots of things. For instante, the free energy for a photon in d-dimensions can be written as follows

    \[\mathcal{F}=-\dfrac{1}{\beta^{d+1}}\dfrac{\Gamma (d+1) \zeta (d+1)}{2^{d-1}\pi^{d/2}\Gamma (d/2) d}\]

and the Casimir energy for a massless electron (OK, in the approximation the electron can be thought as “massless”), reads (in D dimensions)

    \[E(D)=\dfrac{f(D)}{L^{D-1}}\Gamma \left(\dfrac{1-D}{2}\right)\zeta \left(\dfrac{1-D}{2}, \dfrac{1}{2}\right)\]

where \zeta (s, Q) is the Hurwitz zeta function. Now, consider the density

    \[\rho =\dfrac{\omega_d}{(2\pi \hbar)^d}\int_0^\infty \dfrac{p^{d-1}}{z^{-1}e^{\beta H}\pm 1}\]

and where H is the hamiltonian, being H=p^2/2m and \beta=1/k_BT, in the non-relativistic case, but the expresion is completely general. The plus sign corresponds to the so-called Bose-Einstein distributions (bosons), and the minus sign can be associated to the Fermi-Dirac distribution (fermions). The awesome fact is that the two integrals are separately PROPORTIONAL, and the proportionality constant is related to ratios of polylogarithms! I wish I had known this when studied Thermodynamics as undergraduate student! Also, define the thermal wavelength

(3)   \begin{equation*} \lambda_T=\hbar\sqrt{\dfrac{2\pi}{mk_BT}}=\hbar \sqrt{\dfrac{2\pi \beta}{m}} \end{equation*}

This thermal wave-length can be generalized to n-dimensions and dispersion relationship E=ap^s as follows

(4)   \begin{equation*} \boxed{\lambda_T (n)=2\pi \hbar \left(\dfrac{a}{k_B T}\right)^{1/s}\left[\dfrac{\Gamma \left(\frac{n}{2}+1\right)}{\Gamma \left(\frac{n}{s}+1\right)}\right]^{1/n}} \end{equation*}

In fact, if you introduce the grand partition function Q, for the previous hamiltonian, as:

(5)   \begin{equation*} Q=Q_0+\dfrac{gV_n \Gamma \left(\frac{n}{s}+1\right)}{(2\pi \hbar)^n a^{n/s}\Gamma \left(\frac{n}{2}+1\right)}\left(k_BT\right)^{n/s}\begin{cases}Li_{n/s+1}(z), BE\\-Li_{n/s+1}(-z),FD\end{cases} \end{equation*}

The result is fully valid, and thus you can write (now, plug \hbar=1 units for simplicity):

(6)   \begin{equation*} \rho=\dfrac{N}{V}=\mbox{density}=\dfrac{1}{(2\pi)^d}\int_0^\infty\dfrac{\varepsilon^{d/2-1}d\varepsilon}{\exp \left(\beta (\varepsilon-\mu)\right)\pm 1} \end{equation*}

as function of

(7)   \begin{equation*} \boxed{\rho\lambda_T^d=Li_{d/2}(z)} \end{equation*}

provided you define the auxiliar function

    \[\zeta (z)=\begin{cases}z, BED\\ -z, FDD\end{cases}\]

so you get that

    \[\rho=\lambda_T^{-d} Li_{d/2} (z)\]

    \[\rho=-\lambda_T^{-d} Li_{d/2} (-z)\]

can be rewritten in compact form as

(8)   \begin{equation*} \boxed{\rho=sgn(\zeta)\lambda_T^{-d}Li_{d/2}(\zeta)} \end{equation*}

Therefore, the grand partition function becomes

    \[\dfrac{\lambda^2}{V}\ln Q(\zeta)=sgn(\zeta)Li_{d/2+1}(\zeta)\]

and Q_{BED}Q_{FDD}=1. From the polylogarithmic grand partition function you can derive ALL the thermodynamics you want to know. For instance, as

    \[U=-\partial_\beta \ln Q\]

    \[N=\langle N\rangle=z\partial_z \ln Q\]

you can easily find out that

(9)   \begin{equation*} \boxed{\beta P \rho^{-1}=\dfrac{Li_{d/2-1}(\zeta)}{Li_{d/2}(\zeta)}} \end{equation*}

(10)   \begin{equation*} \boxed{\dfrac{\beta U}{N}=\dfrac{d}{2}\dfrac{Li_{d/2-1}(\zeta)}{Li_{d/2}(\zeta)}} \end{equation*}

(11)   \begin{equation*} \boxed{\dfrac{\beta T S}{N}=\left(\dfrac{d}{2}+1\right)\dfrac{Li_{d/2-1}(\zeta)}{Li_{d/2}(\zeta)}\ln \zeta} \end{equation*}

(12)   \begin{equation*} \boxed{Y=yield=\dfrac{Li_{d/2-1}(\zeta)}{Li_{d/2}(\zeta)}} \end{equation*}

Indeed, the proportionality constant between the BED and the FDD as integral in any number of dimensions (e.g., d) is given by

(13)   \begin{equation*} \boxed{K_d=-\dfrac{Li_{d+1}(-1)}{Li_{d+1}(1)}=\left(1-2^{-d}\right)} \end{equation*}

The Riemann zeta function and the polylogarithms help us to write analytical formulae for the Bose-Einstein condensation (the reason for this post). For a non-relativistic boson gas in d dimensions you obtain

(14)   \begin{equation*} \boxed{k_BT_C (NR)=\dfrac{2\pi \hbar^2}{m}\left(\dfrac{n}{\zeta (d/2)}\right)^{2/d}} \end{equation*}

For the 3d case (d=3), you can easily write the common result

    \[k_BT=\dfrac{2\pi \hbar^2}{m\zeta (3/2)}n^{2/3}\approx 3.31\hbar^2 \dfrac{n^{2/3}}{m}\]

where \zeta (3/2)\approx 2.612. Indeed, there are variations of the formulae above. Firstly, M. Casas studied a massless boson gas weakly interacting with dispersion relationship E=a(d)v_F \hbar K, where K is a constant (not related with previous K_d) and v_F is certain Fermi velocity of the bosonic Cooper pair created by two fermions. The condensation formula reads in this case:

(15)   \begin{equation*} \boxed{k_BT=a(d)\hbar v_F\left(\dfrac{\pi^{(d+1)/2}n}{\Gamma\left(\frac{d+1}{2}\right)\zeta (d)}\right)^{1/d}} \end{equation*}

Note that the fact of having a BE condensation depends on the details of the dimensionality of the system and the dispersion relationship. In the end, is the polylogarithm and the Riemann zeta function the main characters there! For instance, in the same non-relativistic case (where before you find out that BEC is impossible in 2d), if you use a confining potential energy U=Ar^\alpha, you can derive the temperature in d-dimensions:

(16)   \begin{equation*} k_BT_C =\left( \left(\dfrac{2\hbar^2}{m}\right)^{d/2}A^{d/2} \dfrac{\Gamma \left(\frac{d}{2}+1\right)n}{\Gamma \left(\frac{d}{\alpha}+1\right)\zeta\left(\frac{d}{2}+\frac{d}{\alpha}\right)}\right)^{\left(\frac{d}{2}+\frac{d}{\alpha}\right)^{-1}} \end{equation*}

and hence, there is only BEC iff

    \[\dfrac{d}{2}+\dfrac{d}{\alpha}>1\]

In the case of the ultra-relativistic case, the condensation formula in d-dimensions reads

(17)   \begin{equation*} \boxed{k_BT_C^{UR}=\left[\dfrac{(\hbar c)^d2^{d-1}\pi^{d/2}\Gamma\left(d/2\right)}{\Gamma (d)\zeta (d)}\right]^{1/d}} \end{equation*}

All these formulae can be even generalized to a fully particle-antiparticle treatment or even take the non-extensive thermostatistics generalization of all of them. More? Yes. Define the biparametric symbols

    \[D_{(q,p)}=\dfrac{f(qx)-f(px)}{(q-p)x}\]

    \[\left[n\right]_{(q,p)}=\dfrac{q^n-p^n}{q-p}\]

such as

    \[D_{(q,p)}x^n=\left[n\right]_{(q,p)}x^{n-1}\]

and the biparametric deformed polylogarithm

    \[Li_{s}^{(q,p)}(z)=\sum_{r=1}^{\infty}\dfrac{\left[n\right]_{(q,p)}z^r}{r^s}\]

Q-ons are the case with symmetrical (q,p)-symbol, so p=1/q. Let me define

    \[H_n(z,k,q)=Li_n(zkq^k)-Li_n(zkq^{-k})\]

and again take E=ap^\sigma. Then, you can derive the magic formula of BEC condensation of q-ons (existing in any dimension!)

(18)   \begin{equation*} \boxed{k_BT_c(q-ons)=\left(\dfrac{\sigma}{2}\right)^{\sigma/d}a(2\pi \hbar)^\sigma 2^\sigma\pi^{\sigma/2}\left(\dfrac{\Gamma (d/2)(q-q^{-1})n}{\Gamma (d/\sigma)H_{d/q+1}(q^2,1,q)}\right)^{\sigma/d}} \end{equation*}

The Stefan-Boltzmann law in D-dimensions (space-like!) can also be related to the zeta function if you write

(19)   \begin{equation*} \boxed{R_T=a(D)T^{D+1}} \end{equation*}

with

(20)   \begin{equation*} \boxed{a(D)=\left(\dfrac{2}{hc}\right)^D\pi^{(D-1)/2}k_B^{D+1}D(D-1)\Gamma\left(\dfrac{D+1}{2}\right)\zeta (D+1)} \end{equation*}

and where the Wien law in D-dimensions (or ND) is related to the Lambert function via

    \[x_D=(D+2)+W(-(d+2)e^{-D-2})\]

Absolutely gorgeous! Isn’t it? See you in another blog post!

 

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