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The paper dealt with quantum canonical ensembles by random walks, where state transitions are triggered by the connections between labels, not by elements, which are transferred. The balance conditions of such walks lead to emission rates of the labels. The labels with emission rates definitely lower than 1 are like modes. For labels with emission rates very close to 1, the quantum numbers are concentrated around a mean value. As an application I consider the role of the zero label in a quantum gas in equilibrium.

In [_{ij}) for the preference of a change from employer i to employer j. I observe the numbers q(i) of employees per employer i during some years. Trying to explain the fluctuations, there are two different models available. If I assume, that always the employees decide to change, the numbers q(i) will follow particle statistics, i.e. they are gaussian like concentrated around a mean value. If I assume, that always the employers decide (without considering anything about employees), the numbers will follow quantum statistics. Then the values of q(i) along such a fluctuation process are similar to a mode ( [_{ij}), where there are many options. Such a process is defined in Chapter 2, with the employees as elements, the employers as labels, and the preference rates as request probabilities.

Normally both models about reasons of fluctuations are mixed. But in statistical physics there is a clean cut. In [

In [

There is an important special case, (dynamic) equilibrium, i.e. detailed balance (Chapter 5). When there is given a positive vector (or function) ρ, there exist transition probabilities for particles for an exchange process in detailed balance ( [

My main reference is [

I started to consider such random walks, trying to explain the difference between Boltzmann and Gibbs entropy more explicitly than in [

Given a directed, connected graph Γ(V,E) with K vertices (labels) V = { 1 , ⋯ , K } , and edges e ∈ E. An edge e leads from label start(e) to label end(e). For labels i, j with i ≠ j, there is at most one edge e ∈ E with i = start(e) and j = end(e). Then I write e = (i → j). The inverse edge is (−e) = (j → i). The graph is assumed to be homogeneous: When e ∈ E, then is −e ∈ E. There is a number L, that for all labels i

# { e ∈ E | s t a r t ( e ) = i } = # { e ∈ E | e n d ( e ) = i } = L (2.1)

Given a function of request probabilities

α : { E → [ 0 , 1 ] e → α e (2.2)

For N > 0 I define random walks through

Q : = { q : V → ℕ 0 | ∑ i = 1 K q ( i ) = N } (2.3)

A single step of the random walk consists of a request and a transition:

request of an edge: (2.4)

select an edge e ∈ R randomly, with same probability for each edge

select a random number τ ∈ [0,1)

if (τ < αe), the request is accepted, otherwise rejected

transition, if the request of edge e = (i → j) is accepted at q ∈ Q: (2.5)

if (q(i) > 0), the transition is accepted and performed by

q → r with r ( i ) = q ( i ) − 1 (annihilation at vertex i) and

r ( j ) = q ( j ) + 1 (creation at vertex j), r(k) = q(k) at k ≠ i,j

if (q(i) = 0), the transition is rejected

On rejection q is not changed.

I call it a quantum process. One version of a corresponding particle process is: I select a particle. It is at label i. Therefore I select an edge starting at i to perform the transition.

Each function α defines a (K x K)-matrix A = (α_{ij}) by

α i j : = { α e for e = ( i → j ) and e ∈ E 0 if there is no e = ( i → j ) ∈ E (2.6)

The result of a finite random walk is summarized by the number of all accepted requests, where q(i) = n for the current state q (i.e. where the transition will start):

c ( i , n ) : = { accepted requests | q ( i ) = n } (3.1)

The probability for a quantum number n at a label i is

p ( q ( i ) = n ) : = c ( i , n ) ∑ m = 0 N c ( i , m ) (3.2)

The mean quantum number at a label (i) is

〈 q ( i ) 〉 : = ∑ n = 0 N n ⋅ c ( i , n ) ∑ n = 0 N c ( i , n ) (3.3)

More typical for a quantum process is the emission rate:

r ( i ) : = p ( q ( i ) > 0 ) = ∑ n = 1 N p ( q ( i ) = n ) (3.4)

i.e. the probability, that a transition starting at label i is accepted, related to all accepted requests starting at label i. The requests are independent of the current state. Therefore counting at all accepted requests leads to the same probabilities.

A special case is N = 1. Then r(i) = p(i), the probability, that the only element is at label i. Particle statistics of N > 1 elements can be explained by N identical and independent systems of such a 1-element system.

The probability of an accepted transition, which ends at a label i (input for i), must be equal to the probability of an accepted transition, which starts at i (output from i). Then, regarding (2.1)

∑ ( e ∈ E , e n d ( e ) = i ) ( α e ⋅ r ( s t a r t ( e ) ) ) = ( ∑ ( e ∈ E , s t a r t ( e ) = i ) ( α e ) ) ⋅ r ( i ) (4.1)

Therefore the values r(i) build an eigenvector of the matrix (2.6), supplied with diagonal elements as in [

α i i : = − ( ∑ ( e ∈ E , s t a r t ( e ) = i ) α e )

As suggested in [

The balance condition (4.1) is a striking property of such systems. In [

Let t(i) : = (number of all accepted transitions starting at label i) ≈ (number of all accepted transitions ending at label i). I define

p ( ( i , n + 1 ) → ( i , n ) ) : = c ( ( i , n + 1 ) → ( i , n ) ) t (i)

p ( ( i , n ) → ( i , n + 1 ) ) : = c ( ( i , n ) → ( i , n + 1 ) ) t (i)

Then there is another balance condition, which is used in a similar context by Einstein to derive Planck’s radiation law ( [

For a label i and a quantum number n there is

p ( ( i , n ) → ( i , n + 1 ) ) = p ( ( i , n + 1 ) → ( i , n ) ) (4.2)

I search for a relation to modes in my context. The probability of an accepted request of edge e in all accepted requests is

p ( requestof e ) = α e ∑ ( b ∈ E ) α b

The probability, that an accepted request starts at label i with quantum number n related to all accepted requests, is

p ( i , n ) = ∑ ( s t a r t ( e ) = j ) α e ∑ ( b ∈ E ) α b

The probability of a rejection of a transition i → j depends on the actual quantum number at the end of the edge, i.e. q(j), because a higher value of q(j) gives a higher probability of the rejection condition “q(i) = zero” at the start of the edge, label i. I have probabilities like “p(q(start(e)) = m) & q(end(e)) = n)”. If I assume independence, I get a relation:

p ( q ( s t a r t ( e ) = m ) & q ( e n d ( e ) ) = n ) = p ( q ( s t a r t ( e ) = m ) ) ⋅ p ( q ( e n d ( e ) ) = n )

(4.3)

Then I get via (3.4)

p ( ( i , n ) → ( i , n + 1 ) ) = ∑ ( e n d ( e ) = i ) α e ⋅ r ( s t a r t ( e ) ) ⋅ p ( q ( i ) = n ) ∑ ( b ∈ E ) α b

p ( ( i , n + 1 ) → ( i , n ) ) = ∑ ( s t a r t ( e ) = i ) α e ⋅ p ( q ( i ) = n + 1 ) ∑ ( b ∈ E ) α b

The balance (4.2) is

∑ ( e n d ( e ) = i ) α e ⋅ r ( s t a r t ( e ) ) ⋅ p ( q ( i ) = n ) = ∑ ( s t a r t ( e ) = i ) α e ⋅ p ( q ( i ) = n + 1 )

p ( q ( i ) = n + 1 ) p ( q ( i ) = n ) = ∑ ( e n d ( e ) = i ) α e ⋅ r ( s t a r t ( e ) ) ∑ ( s t a r t ( e ) = i ) α e

Therefore the quotient is independent of the quantum number n. Such a constant quotient, if < 1, describes the distribution of the quantum numbers of a mode ( [

〈 q ( i ) 〉 = r ( i ) 1 − r ( i ) = 1 1 r ( i ) − 1 (4.4)

In the counting results of my random walks (4.4) the values are nearly equal, excepted when the value of r(i) is close to 1.

Given a vector

ρ : V → ℝ + (5.1)

I define “equilibrium” by request probabilities for each edge e = ( i → j ) ∈ E by

α ( i → j ) : = min { 1 , ρ ( j ) ρ ( i ) } (5.2)

as usual in the Metropolis algorithm or via log ρ instead of ρ in the Hybrid Monte Carlo algorithm [

α ( i → j ) ⋅ ρ ( i ) = min { ρ ( i ) , ρ ( j ) } = α ( j → i ) ⋅ ρ ( j ) (5.3)

Therefore especially the balance conditions (4.1) are fulfilled by the vector ρ. The equilibrium for ρ is independent of the set of edges E, which is used for the transitions. Only (5.2) and irreducibility of the matrix (2.6) is required (for (2.1) one adds edges with request probabilities 0). In the context of the Metropolis-Hastings algorithm [

Such balance conditions are not available in the boson system of [_{p} = N) is eliminated due to the passage to the grand partition function ( [

ρ : { Q → ℝ + q → ∏ i = 1 K ρ ( i ) q ( i ) (5.4)

The request probability of an edge (q → r) for q, r ∈ Q is

α ( q → r ) : = { α ( i → j ) when there is an edge ( i → j ) ∈ E with q = r , excepted q ( i ) − 1 = r ( i ) and q ( j ) + 1 = r ( j ) 0 otherwise

I can fulfill condition (2.1) e.g. adding multiple edges (q → q).

The extended vector ρ fulfills the corresponding detailed balance conditions (5.3). As mentioned before, because of N = 1, there is

r ( q ) = p ( q ) = probability of occurrences of q in the random walk (5.5)

Now N may vary. I write Q(N) for Q defined by (2.3), p_{N} instead of p and r_{N} instead of r. Then with Z 0 : = 1 and

Z N : = ∑ ( q ∈ Q ( N ) ) ρ ( q ) = the complete homogeneous symmetric polynomial in K variables ρ ( i ) of degree N

p N ( q ) = ρ ( q ) Z N ( probabilities have sum = 1 )

p N ( q ( i ) = n ) = ∑ ( q ∈ Q ( N ) , q ( i ) = n ) ρ ( q ) Z N

r N ( i ) = p N ( q ( i ) > 0 ) = ∑ ( q ∈ Q ( N ) , q ( i ) > 0 ) p ( q ) = ∑ ( q ∈ Q ( N ) , q ( i ) > 0 ) ρ ( q ) Z N = ρ ( i ) ⋅ Z N − 1 Z N ( using ( 5.4 ) ) (5.6)

r N ( i ) r 1 ( i ) = Z 1 ⋅ Z N − 1 Z N ( independent of i ) (5.7)

It confirms, that for each N the emission rates are multiples of the same eigenvector, and that N = 1 leads to probabilities. Furthermore r_{N}(i) increases with N and the limit for

N → ∞ is 1 (proof via Z-functions). Because of (5.6) the emission rates (more general: all probabilities of quantum numbers) are independent of variations of the edges and request probabilities as mentioned at (5.3), detailed balance. Writing ρ ( i ) = exp ( − β H ( i ) ) I get Z-functions as usual.

Normally the labels are like modes, but there are exceptions. If ρ(i) is maximal at label i, there are two significant types. The label may be like a mode too (typically at low values of β or low values of N), or its quantum numbers may be concentrated, i.e. nearly gaussian distributed around a mean value (typically at large values of β or large values of N). There are continuous passages between these types.

^{10} steps. Evaluations start at 1/2 of all steps, to achieve a randomized start position.

Now I look at special functions ρ. In the context of quantum gases I find a