^{1}

^{*}

^{2}

^{3}

^{4}

In many interesting physical examples, the partition function is divergent, as first pointed out in 1924 by Fermi (for the hydrogen-atom case). Thus, the usual toolbox of statistical mechanics becomes unavailable, notwithstanding the well-known fact that the pertinent system may appear to be in a thermal steady state. We tackle and overcome these difficulties hereby appeal to firmly established but not too well-known mathematical recipes and obtain finite values for a typical divergent partition function, that of a Brownian particle in an external field. This allows not only for calculating thermodynamic observables of interest, but for also instantiating other kinds of statistical mechanics’ novelties.

In many interesting physical examples, the partition function is divergent [

We will consider here the partition function for Brownian motion in an external field, given by [

Z = ∫ − ∞ ∞ e β U 0 1 + x 2 d x , (2.1)

with β = 1 / ( k B T ) and k B = Boltzmann’s constant. Change now variables to y = 1 + x 2 . Taking advantage now of well-known features of Schwartz’ theory of distributions [

Z = ∫ 1 ∞ ( y − 1 ) − 1 2 e β U 0 y d y ≡ lim ν → 1 ∫ 1 ∞ y ν − 1 ( y − 1 ) − 1 2 e β U 0 y d y , (2.2)

and remember that the limit of an integral equals the integral of the limit. We consult then the Tableof Ref. [

W = ∫ u ∞ x ν − 1 ( x − u ) μ − 1 e β x d x = B ( 1 − μ − ν , μ ) u μ + ν − 1 ϕ ( 1 − μ − ν ; 1 − ν , β u ) . (2.3)

Here B is the well-known beta function and ϕ the confluent hypergeometric function, that reads, appealing to the Gamma function Γ ,

B = Γ ( 1 − μ − ν ) Γ ( μ ) / Γ ( 1 − ν ) . (2.4)

Comparing integrals, we see at this stage that the right hand side of (2.2) will coincide with W in (2.3) by setting

μ = 1 / 2 ; ν = 1 ; u = 1 , (2.5)

so that these special values are to be inserted in

W = Γ ( 1 − μ − ν ) [ Γ ( μ ) / Γ ( 1 − ν ) ] u μ + ν − 1 ϕ ( 1 − μ − ν ; 1 − ν , β u ) . (2.6)

Note also that

Γ ( 1 / 2 ) = π ; Γ ( − 1 / 2 ) = − 2 π . (2.7)

We have a Γ ( 0 ) in a denominator now. This induces us to appeal once again to [

lim γ → 0 ϕ ( α ; γ ; s ) = z α ϕ ( α + 1 ; 2 ; z ) , (2.8)

so that we can finally arrive at the result

Z = π β U 0 ϕ ( 1 2 ; 2 ; β U 0 ) , (2.9)

our desired finite form. We see that we arrive at Z via a straightforward path. The essential step here is that of consulting an appropriate table of integrals and performing adequate manipulations. Note that at very low temperatures quantum effects raise their head and our treatment becomes invalid. Below it will be shown that one also encounters problems or exceedingly high temperatures. We have found a finite partition function for our Brownian problem and proceed to calculate with it, below, important quantifiers of statistical mechanics.

We find it convenient to plot our thermal quantities versus y = k B T / U 0 in the range 0 ≤ y ≤ 1 . Given the smallness of k B , this encompasses an immense T-range, since k B is of the order of 10^{−}^{23} in its appropriate units. In particular, we plot the logarithm of the partition function in

One has

〈 U 〉 = − ∂ ln Z ∂ β , (3.1)

so that

〈 U 〉 = − 1 Z [ π U 0 ϕ ( 1 2 ; 2 ; β U 0 ) + π β U 0 2 4 ϕ ( 3 2 ; 3 ; β U 0 ) ] . (3.2)

Note that at very low temperatures quantum effects raise their head and our classical treatment becomes invalid.

We have

S = ∂ ( k B T ln Z ) ∂ T , (3.3)

so that

S = ln [ π β U 0 ϕ ( 1 2 ; 2 ; β U 0 ) ] − β Z [ π U 0 ϕ ( 1 2 ; 2 ; β U 0 ) + π β U 0 2 4 ϕ ( 3 2 ; 3 ; β U 0 ) ] , (3.4)

that is plotted in

One defines it as

C = − β T ∂ 〈 U 〉 ∂ β , (3.5)

so that

C = − 1 Z 2 [ π U 0 ϕ ( 1 2 ; 2 ; β U 0 ) + π β U 0 2 4 ϕ ( 3 2 ; 3 ; β U 0 ) ] × [ π β U 0 T ϕ ( 1 2 ; 2 ; β U 0 ) + π β 2 U 0 2 4 T ϕ ( 3 2 ; 3 ; β U 0 ) ] + 1 Z [ π β U 0 2 2 T ϕ ( 3 2 ; 3 ; β U 0 ) + π β 2 U 0 3 8 T ϕ ( 5 2 ; 4 ; β U 0 ) ] , (3.6)

depicted in

We pass to the moment generating function for our extant probability distribution function (PDF) f ( x ) [consult (2.1)]

f ( x ) = e β U 0 1 + x 2 Z , (4.1)

where Z is given by (2.1). In the naive traditional treatment, these moments diverge. The mean value for x 2 n + 1 , ( n = 1 , 2 , 3 , ⋯ ) vanishes by parity. That of x 2 n becomes

〈 x 2 n 〉 = 1 Z ∫ − ∞ ∞ x 2 n e β U 0 1 + x 2 d x . (4.2)

Appeal again to the variables change y = 1 + x 2 and face

〈 x 2 n 〉 = 1 Z ∫ 1 ∞ ( y − 1 ) n − 1 2 e β U 0 y d y , (4.3)

so that, proceeding in a fashion similar to that above we find

〈 x 2 n 〉 = β U 0 Z Γ ( − n + 1 2 ) Γ ( n + 1 2 ) ϕ ( 1 2 − n ; 2 ; β U 0 ) , (4.4)

Thus, we get for the moment generating function M 1 ( t )

M 1 ( t ) = β U 0 Z ∑ n = 0 ∞ t 2 n ( 2 n ) ! Γ ( 1 2 − n ) Γ ( 1 2 + n ) ϕ ( 1 2 − n ; 2 ; β U 0 ) . (4.5)

As particular cases, we obtain the values

〈 x 2 〉 = − π β U 0 Z ϕ ( − 1 2 ; 2 ; β U 0 ) , (4.6)

and

〈 x 4 〉 = π β U 0 Z ϕ ( − 3 2 ; 2 ; β U 0 ) . (4.7)

The first one is plotted in ^{22} Kelvin. For reference, 100 seconds after the Big Bang it is estimated that the temperature is of a billion K-degrees, and 0.0001 seconds after the Big Bang it is of about T = 10^{13} K [

Given a continuous probability distribution function (PDF) f ( x ) with x ∈ Δ ⊂ ℝ and ∫ Δ f ( x ) d x = 1 , its associated Shannon Entropy S is, as we saw above,

S ( f ) = − ∫ Δ f ln ( f ) d x (5.1)

a quantifier of global nature that it is not very sensitive to strong changes in the distribution that may take place in a small-sized region. This is not the case for Fisher’s Information Measure (FIM) F [

F ( f ) = ∫ Δ 1 f ( x ) [ d f ( x ) d x ] 2 d x = 4 ∫ Δ [ d ψ ( x ) d x ] 2 (5.2)

FIM can be interpreted in variegated fashions. 1) As a quantifier of the ability

to estimate a parameter. 2) As the amount of information that can be extracted from a set of measurements. 3) A quantifier of the state of disorder of a system or phenomenon [

For the f of (3.5) one has

F ( f ) = 2 Z ∫ 0 ∞ e β U 0 1 + x 2 [ de β U 0 1 + x 2 d x ] 2 d x , (5.3)

or

F ( f ) = 8 β 2 U 0 2 Z ∫ 0 ∞ x 2 ( 1 + x 2 ) 2 e β U 0 1 + x 2 d x . (5.4)

Changing variables in the fashion y = 1 + x 2 we get

F ( f ) = 4 β 2 U 0 2 Z ∫ 1 ∞ y − 2 ( y − 1 ) 1 2 e β U 0 y d y , (5.5)

that after evaluation yields for the Fisher information measure the value

F ( f ) = 2 β U 0 , (5.6)

clearly a very large positive number, given the smallness of the Boltzmann constant entering the denominator. Let us look for the Cramer-Rao (CR) product [ F ( f ) 〈 x 2 〉 f ] , that is always ≥ 1 [

We need a value for 〈 x 2 〉 , that we take from (4.4). The Cramer-Rao product 〈 x 2 〉 F is then

F ( f ) 〈 x 2 〉 = − 2 π β 2 U 0 2 Z ϕ ( − 1 2 ; 2 ; β U 0 ) . (5.7)

The CR product is plotted in ^{22} Kelvin, at which our probability distribution no longer makes sense. We have already encountered above this effect, in connection with 〈 x 2 〉 -graph, the entropy, and the specific heat.

In deceptively simple fashion, we have regularized the partition function for Brownian functions moving in an external potential, thus solving a very old

problem. Some other special cases were already treated by the present authors. One is that of the Z-expression in the case of Newton’s gravity [

· Being of a classical nature, it fails at very low temperatures, where quantum effects become predominant.

· At extremely high temperatures, of the order of 10^{22} Kelvin, we face a T-upper bound. This fact has already been reported, in another context, by Refs. [

Summing up: We were here tackling partition function’ divergences, a physically-motivated mathematical problem, that we indeed solved. As for applications, the most we can say at this stage is that we have at our disposal a new canonical probability distribution. Can one use the concomitant partition function Z in a concrete problem? To answer this question, more research is needed. We guess that with this Z some density distribution might be constructed that could describe a quasi-stationary solution in some suitable scenario.

The authors declare no conflicts of interest regarding the publication of this paper.

Plastino, A., Rocca, M.C., Monteoliva, D. and Hernando, A. (2021) Brownian Motion in an External Field Revisited. Journal of Modern Physics, 12, 82-90. https://doi.org/10.4236/jmp.2021.122008