Journal of Modern Physics, 2018, 9, 1448-1458
http://www.scirp.org/journal/jmp
ISSN Online: 2153-120X
ISSN Print: 2153-1196
Quantum Statistics of Random Walks
Manfred Harringer
Independent Researcher, Cologne, Germany
How to cite this paper: Harringer, M.
(2018) Quantum Statistics of Random Walks.
Journal of Modern Physics, 9, 1448-1458.
https://doi.org/10.4236/jmp.2018.97089
Received: May 16, 2018
Accepted: June 24, 2018
Published: June 27, 2018
Copyright © 2018 by author and
Scientific Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
Abstract
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.
Keywords
Random Walks, Particle Statistics, Boson Statistics, Balance Conditions,
Detailed Balance, Quantum Gas, Perron-Frobenius Theory
1. Introduction
In [1] quantum statistics starts with the grand canonical ensemble. The quantum
canonical ensemble is mentioned, but not elaborated. I want to fill this gap.
There is a simple example, which corresponds to a quantum canonical ensemble:
There are K employers and N employees. I want to describe the fluctuation of
employees between the employers. I assume, that there are rates (αij) 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 ([2] p. 100): for a mode i there is a
value r, where p ( q ( i ) = n ) = (1 − r ) ∗ r n ), excepted for the values of the employer,
DOI: 10.4236/jmp.2018.97089 Jun. 27, 2018
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M. Harringer
who is most preferred by the change rates (αij), 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 [1] Huang introduces different kinds of
elements: particles, bosons and fermions. Then particles are treated in particle
statistics, bosons and fermions in quantum statistics. I search for such differences
elsewhere, in the trigger method of state transitions. In my quantum systems it is
possible, that a request for a transition is rejected, because there is no element
available to perform the transition. In my example above it is artificial. But it is
essential, when I choose such a model.
In [3] states and state changes are described by transition probabilities of
complexes. I consider most simple complexes, i.e. single exchanges between
species (labels). In [3] the number of particles per species (label) is observed along
several steps of a transition process. There are balance conditions for the transition
process (in a special case, [3] 16.3). Instead of transition probabilities for particles I
build a quantum analogue by request probabilities (Chapter 2) with nearly the
same balance conditions (Chapter 4). The common feature is a unique eigenvector
(up to a factor λ > 0), unique because of the theorem of Perron-Frobenius ([3],
Chapter 16.6). In [3] the eigenvector consists of probabilities with sum = 1. In the
corresponding quantum system it consists of emission rates, where the highest
emission rate has a value between 1/K and 1.
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
([4] Metropolis algorithm, or with ρ =
( x ) exp ( − β H ( x ) ) in the hybrid Monte
Carlo method). I use the same values as request probabilities. The eigenvector ρ
is the same for all numbers N of elements of the exchange process. When N = 1,
transition probabilities and request probabilities coincide. When N increases, the
emission rates increase by a common factor. Another setup is: there is a space X,
where I can build approximately equal distributed finite sets, with a function
H : X → . Then the selection of these finite sets varies, and one asks for
properties of the exchange processes, which are independent of the selection.
Assuming equilibrium, the determination of the edges of the state transitions is
less important than it is e.g. in models of equilibration as in [5] for quantum
systems, or in approximation tasks by the Metropolis-Hastings algorithm [4].
My main reference is [3] (which mentions many additional references)
because of the balance conditions and the eigenvector. Then I search for suitable
labels to count elements. The particle systems in [3] are not related to any
mechanical particle motions. Therefore the labels must not be related to the
moving objects of quantum mechanics, and it is not required, that single steps
are unitary transformations as in usual quantum random walks [6]. Instead of
ensembles ([4] Chapter 10.1) I observe the routes of random walks to derive
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probabilities and mean values. For connections to statistical physics I use [1] or
[4] as main reference for an ideal quantum gas and the black box radiation.
There I find suitable labels (Chapter 6).
I started to consider such random walks, trying to explain the difference
between Boltzmann and Gibbs entropy more explicitly than in [7]. It was my
“Gibbs version” of the random walks, which led to my version of quantum
random walks. I made numerous computer simulations, observing the results of
such random walks, to confirm my theoretical considerations.
2. Random Walks
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 | start ( e ) =
i} =
# {e ∈ E | end ( e ) =
i} =
L
(2.1)
Given a function of request probabilities
E → [ 0,1]
e → α e
α :
(2.2)
For N > 0 I define random walks through
{
}
Q :=
q : V → 0 | ∑ i =1 q ( i ) =
N
K
(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 = (αij) by
α e for e =
( i → j ) and e ∈ E
α ij :=
0 if there is no e = ( i → j ) ∈ E
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(2.6)
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M. Harringer
3. Evaluations
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
c ( i, n )
p ( q (=
i ) n=
):
(3.2)
∑ m=0c ( i, m )
N
The mean quantum number at a label (i) is
∑ n ⋅ c ( i, n )
q ( i ) := n=N0
∑ n =0 c ( i , n )
N
(3.3)
More typical for a quantum process is the emission rate:
r (=
i ) : p ( q ( i ) >=
0)
N
(i ) n)
∑ n=1 p ( q=
(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.
4. Balance Conditions
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 ,end ( e ) i )
( e))) ( ∑ (
(α ⋅ r ( start =
e
=
e∈E , start ( 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 [3]
α ii := −
(∑(
α
)
e∈E , start ( e )=
i) e
As suggested in [3], I can add a common value λ > 0 to the diagonal elements,
to achieve non negative matrix elements. The matrix must be irreducible. Then
the values r(i) build an exemplar of the unique eigenvector with positive
components due to the theorem of Perron-Frobenius.
The balance condition (4.1) is a striking property of such systems. In [3] it is
considered only for particle statistics.
Let t(i) := (number of all accepted transitions starting at label i) ≈ (number
of all accepted transitions ending at label i). I define
c ( ( i, n + 1) → ( i, n ) )
p ( ( i, n + 1) → ( i, n ) ) :=
t (i )
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M. Harringer
c ( ( i, n ) → ( i, n + 1) )
p ( ( 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 ([8], and there are many related
presentations available).
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 ( request of e ) =
αe
∑ (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 ) =
∑ ( start(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 ( start ( e ) =
m ) & q ( end ( e ) ) =
n =
p q ( start ( e ) =
m ) ⋅ p q ( end ( e ) ) =
n
(4.3)
Then I get via (3.4)
n)
∑ (end (e)=i )α e ⋅ r ( start ( e ) ) ⋅ p ( q ( i ) =
∑ (b∈E )α b
p ( ( i, n ) → ( i, n + 1) ) =
n + 1)
∑ ( start(e)=i )α e ⋅ p ( q ( i ) =
∑ (b∈E )α b
p ( ( i, n + 1) → ( i, n ) ) =
The balance (4.2) is
∑
α ⋅ r ( start ( e ) ) ⋅ p ( q ( i ) =
n) =
∑
e
=
( end ( e ) i )=
( start ( e ) i )
p ( q ( i )= n + 1)
p ( q (i ) = n)
=
αe ⋅ p ( q (i ) =
n + 1)
∑ (end (e)=i )α e ⋅ r ( start ( e ) )
∑ ( start(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 ([2] p. 100). Therefore I can try to compare the labels with modes. For a
perfect mode at label i there would be
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M. Harringer
q (i )
=
r (i )
=
1 − r (i )
1
1
−1
r (i )
(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.
5. Equilibrium
Given a vector
ρ :V → +
(5.1)
I define “equilibrium” by request probabilities for each edge e = ( i → j ) ∈ E
by
ρ ( j )
ρ ( i )
α ( i→ j ) := min 1,
(5.2)
as usual in the Metropolis algorithm or via log ρ instead of ρ in the Hybrid
Monte Carlo algorithm [4] for transition probabilities. The vector ρ fulfills the
detailed balance conditions, because for each pair ((i → j), (j → i)) of edges there
is
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 [4] there are additional options to vary.
Such balance conditions are not available in the boson system of [1] Chapter
8.5, which consists of a set of independent labels p, because there the
precondition ([1] (8.57) Σnp = N) is eliminated due to the passage to the grand
partition function ([1] (8.61)). It leads to independent labels and balance
conditions like (4.2) at each single label p. Along a random walk, the edge
selection is independent of the current state. Therefore I can interpret the
random walk through Q as a random walk with Q as set of labels of a single
element (K = #Q, N = 1), extending ρ :V → + to
Q → +
K
q( i )
q → ∏ i=1 ρ ( i )
ρ :
(5.4)
The request probability of an edge (q → r) for q, r ∈ Q is
r,
α ( i→ j ) when there is an edge ( i → j ) ∈ E with q =
=
=
=
α ( 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
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M. Harringer
(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), pN instead of p and rN
instead of r. Then with Z 0 := 1 and
=
ZN :
∑=
( q∈Q ( N ) ) ρ ( q )
the complete homogeneous symmetric polynomial
in K variables ρ ( i ) of degree N
pN ( q )
=
ρ (q)
=
( probabilities have sum 1)
ZN
pN ( q ( i=
) n=)
ρ (q)
∑ ( q∈Q( N ),q(i )=
n)
Z
N
=
rN ( i ) pN ( q ( i ) > 0 )
= ∑ ( q∈Q( N ),q( i )>0 ) p ( q )
= ∑ ( q∈Q( N ),q( i )>0 )
= ρ (i ) ⋅
Z N −1
ZN
ρ (q)
(5.6)
ZN
( using ( 5.4 ) )
rN ( i )
Z
= Z1 ⋅ N −1 ( independent of i )
r1 ( i )
ZN
(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 rN(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.
6. Applications
6.1 Main Results
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.
Table 1 contains counting results and derived results for K = 12 labels with
values H(i), randomly distributed in [0,1], labels sorted by the H-values, β = 1,
N = 80. The random walk consists of 1010 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
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Table 1. Emission rates and mean values for random values of H(i).
label i
H(i)
r( i)
t (i)
< q ( i)>
< r-mode( i)> < t-mode( i)>
1
0.042093
0.997428
0.997384
43.576472
387.742031
381.324757
2
0.117060
0.925058
0.925348
11.658032
12.343617
12.395400
3
0.121573
0.920967
0.921181
11.109490
11.652974
11.687266
4
0.207110
0.845804
0.845661
5.426121
5.485244
5.479257
5
0.506089
0.626902
0.627121
1.678370
1.680264
1.681837
6
0.694566
0.519745
0.519394
1.080761
1.082228
1.080706
7
0.706656
0.512974
0.513153
1.054312
1.053278
1.054032
8
0.754032
0.489525
0.489408
0.958108
0.958959
0.958512
9
0.766346
0.483657
0.483419
0.936427
0.936696
0.935804
10
0.813673
0.461302
0.461073
0.855559
0.856327
0.855538
11
0.819835
0.458266
0.458241
0.845412
0.845926
0.845838
12
0.836762
0.450872
0.450549
0.820937
0.821070
0.819999
r(i) = emission rate of label i by the counting results c(i, n); t(i) = emission rate of label i by Z-functions (5.6)
(“theoretical”); <q(i)> = mean quantum number at label i by the counting results c(i, n); <r-mode(i)> =
r(i)/(1 − r(i)) = mean quantum number of the corresponding mode <t-mode(i)> = t(i)/(1 − t(i)).
basic relation of Bose statistics in [9] (11.4):
nk =
1
(6.1)
exp ( β ( ε k − µ ) ) − 1
It means: A vector as required in (5.1) is
ρ=
( k ) exp ( − βε k )
I assume Equilibrium (5.2) of a quantum process. Then ρ(k) becomes an
eigenvector, because of the detailed balance (5.3). The emission rates build an
eigenvector too. I assume, that there is a unique maximum of ρ in k = 1 (i.e. ε1 is
minimal). Then there is a unique μ < ε1 with r (1) :=exp ( − β ( ε1 − µ ) ) , and
therefore for each label k
r ( k ) = exp ( − β ( ε k − µ ) )
(6.2)
Then (6.1) coincides with (4.4), which is well approximated in my random
walks for labels k > 1. But the label k = 1 is an exception, not for (6.2), but for
(6.1). Therefore k = 1 is treated by an extra term in [1] and [4].
6.2. Ideal Bose gas
If I assume detailed balance of the exchange of kinetic energy in an ideal Bose
gas [1] or [4], I have 3D vectors p with
εp =
p
2
2m
2
p
ρ β ( p=
) exp − β ⋅
2m
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In such systems there exist critical temperatures, where the label (p = 0) is no
longer like a mode (Bose-Einstein condensation). One finds a critical
temperature on this way [10]: I build families of systems by their Hamilton
operator ([10] (1.4)). β is fixed. I calculate the value of the emission rate r(p = 0)
(called fugacity in this context) for single members of the family. Then I build a
limit of the systems, running to infinity (there it is L → ∞). Although r(p = 0) < 1
for each single system, the limit may be 1. Then β varies, and the critical value
βc = 1/kTc is the minimal β (maximal temperature), where the limit is 1. Then I
can show, that in the limit the zero label (p = 0) has too many elements in mean,
to be like a mode. It remains valid to claim, that for finite systems there are
continuous passages between the different types of behavior at the label (p = 0).
The effect of Bose-Einstein condensation is observed in [11] based upon a
birth and death process. It contains steps modeled by single transitions from one
species to another. I guess, that there exist steps with rejections because of an
empty resource. But I cannot find it.
6.3. Photon Gas
If I assume detailed balance of the exchange of the photon energy in an ideal
photon gas ([1] Chapter 12.1), I have
ρ β ( k=
) exp ( − β ⋅ ωk )
(6.4)
with ωk := c k and Planck’s constant .
The label k is a 3D-vector. The setup for the random walks is similar as in
(6.2). But here instead of (6.1) there is in [1] (12.8)
nk =
2
exp ( β ⋅ ωk ) − 1
(6.5)
“2” indicates, that there are two labels for each vector k, because of two
different polarizations. There are several options to determine the labels: When
there are two labels, I am asked about the request probabilities for transitions
between these labels (which are = 1, because here the ωk-values are equal). Or I
assume one common label for both polarizations, because the exchange is
independent of the polarization. Or there is no exchange between different
polarizations and therefore I have two independent systems of the same kind.
Apart from the factor 2 it is just black box radiation. The idea behind (6.5) is,
that in this case all labels are like modes. But there exist experimental results
about special photon gases, where there is an exceptional zero label [12]. In [13]
it is explained assuming a grand canonical ensemble in the case of modes and a
canonical ensemble in the exceptional case.
Table 2 contains counting results and derived results for a photon gas model,
which includes both cases. The setup is: H(x) = |x|, a 3D ball of radius 6 with K =
2000 labels equal distributed in the ball, 0 is included, β = 0.9. It leads to maximal
energies per value |x| at |x| ≈ 3.0 … 3.3. Each random walk consists of 1010 steps.
Evaluations start at 1/2 of all steps. The number of elements N is varied.
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Table 2. Emission rates and mean values for a finite model of a photon gas.
N
r(0)
r( y)
< q (0)>
< q ( y)>
< mode(0)>
< mode( y)>
zeroes
60
0.713194
0.507569
2.288819
1.024469
2.486677
1.030741
0
80
0.901743
0.628189
6.458483
1.652551
9.177413
1.689536
0
100
0.978373
0.694037
14.391452
2.237323
45.238498
2.268371
0
120
0.998734
0.709060
31.147484
2.430359
789.014220
2.437137
0
140
1.000000
0.707658
54.831132
2.394297
none
2.420653
9
160
1.000000
0.705996
73.949548
2.400769
none
2.401319
32
180
1.000000
0.705799
92.366046
2.385630
none
2.399034
46
“0” is the zero label; label y is the nearest neighbor of the zero label; r(i) = emission rate of label i by the
counting results c(i, n); <q(i)> = mean quantum number at label i by the counting results c(i, n); <mode(i)>
= r(i)/(1 − r(i)) = mean quantum number of the corresponding mode; zeroes = max {n|c(0, ν) = 0 for each ν
< n.}
At N = 60 label 0 is like a mode. Then there is a continuous passage, and
beginning at N = 120, label 0 becomes concentrated around a mean value and
r(y) no longer increases. Although the random walks are rather long, the values
have random deviations, because K is large. As a counting result, I get r(0) = 1.
The theoretical value remains < 1. The values of the corresponding Z-functions
in (5.6) are around 1031.
7. Conclusions
I considered 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 cannot be
exactly like modes, in contrast to [2] (12.2). They are not connected with
electromagnetic waves and their superposition principle. But I can build models
of dynamical equilibrium, which gain insight into the statistics of the quantum
numbers, and I can try to compare it e.g. with a Bose-Einstein condensate. There
is no singularity, because the zero label is not introduced as a mode with mean
quantum number, but as a label with an emission rate. To get comparable
quantitative results there is an important question left open: How are the
numbers of elements and labels related to pressure and volume of a quantum
gas?
Another question is: Are there systems, where the zero label of my finite
quantum system can explain the properties of the vacuum of a corresponding
quantum system?
Comparisons are difficult, because I considered equilibrium only. It is left
open, how to introduce a quantitative notion of time.
References
DOI: 10.4236/jmp.2018.97089
[1]
Huang, K. (1987) Statistical Mechanics. 2nd Edition, John Wiley & Sons, New York.
[2]
Kittel, C. and Kroemer, H. (1989) Physik der Wärme. 3rd Edition, R. Oldenbourg,
1457
Journal of Modern Physics
M. Harringer
Munic, Oldenburg.
[3]
Baez, J.C. and Biamonte, J.D. (2012) A Course on Quantum Techniques for Stochastic Mechanics. arXiv:1209.3632v1 [quant-ph].
[4]
Tuckerman, M.E. (2010) Statistical Mechanics: Theory and Molecular Simulation.
Oxford University Press, Oxford.
[5]
Wolschin, G. (2018) Physica A, 499, 1-10.
https://doi.org/10.1016/j.physa.2018.01.035
[6]
Kempe, J. (2003) Quantum Random Walks—An Introductory Overview. arXiv:quant-ph/0303081v1.
[7]
Miranda, E.N. (2015) Journal of Modern Physics, 6, 1051-1057.
https://doi.org/10.4236/jmp.2015.68109
[8]
Einstein, A. (1917) On the Quantum Theory of Radiation. Physikalische Zeitschrift
Bd. 18.
[9]
Plischke, M. and Bergerson, B. (2005) Equilibrium Statistical Physics. 3rd Edition,
World Scientific Publishing, Hackensack.
[10] Davies, E.B. (1972) Communications in Mathematical Physics, 28, 69-86.
https://doi.org/10.1007/BF02099372
[11] Bianconi, G., Ferretti, L. and Franz, S. (2009) Non-Neutral Theory of Biodiversity.
Europhysics Letters, 87, P07028. https://doi.org/10.1209/0295-5075/87/28001
[12] Damm, T., Schmitt, J., Qi Liang, D. Dung, F. Vewinger, M. Weitz and J. Klaers,
(2016) Nature Communications, 7, Article Number: 11340.
https://doi.org/10.1038/ncomms11340
[13] Klaers, J., Schmitt, J., Vewinger, F. and Weitz, M. (2010) Nature, 468, 545-548.
DOI: 10.4236/jmp.2018.97089
1458
Journal of Modern Physics