Stochastic Diagrams for Critical Point Spectra

57 Pages·0.54 MB·English

by  S. Chaturvedi

Stochastic Diagrams for Critical Point Spectra
8 S. Chaturvedi∗ and P. D. Drummond
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9 Department of Physics, University of Queensland
1 St Lucia 4067, Queensland, Australia.

n
a February 1, 2008
J

9

1
Abstract
v
2 A new technique for calculating the time-evolution, correlations and
7
steady state spectra for nonlinear stochastic differential equations is pre-
0
sented. To illustrate the method, we consider examples involving cubic
1
nonlinearities in an N-dimensional phase-space. These serve as a useful
0
paradigmfordescribingcriticalpointphasetransitionsinnumerousequi-
8
9 librium and non-equilibrium systems. The technique presented here is
/ not perturbative. It consists in developing the stochastic variable as a
t
a power series in time, and using this to compute the short time expan-
m sion for the correlation functions. This, in turn, is extrapolated to large
times and Fourier transformed to obtain thespectrum. A stochastic dia-
-
d gram technique is developed to facilitate computation of the coefficients
n of the relevant power series expansion. Two different types of long-time
o extrapolation technique, involvingeither simple exponentials or logarith-
c micrationalapproximations,areevaluatedforthird-orderdiagrams. The
:
v analyticalresultsthusobtainedarecomparedwithnumericalsimulations,
i together with exact results available in special cases. The agreement is
X
foundtobeexcellent uptoandin theneighborhood of thecritical point.
r Theexponential extrapolation works especially well even abovethecriti-
a
cal point at large N-values, where the dynamics is one of phase-diffusion
in the presence of a spontaneously broken symmetry. A feature of this
method is that it also enables the calculation of the steady state spectra
ofpolynomialfunctionsofthestochasticvariable. Inthesecases,thefinal
correlations can be non-bistable even above threshold, and the logarith-
micrationalextrapolationhasthegreateraccuracy. Finally,weemphasize
that the techniqueis also applicable to more general stochastic problems
involvingspatial variation in addition to temporal variation.
∗Permanent address : School of Physics, University of Hyderabad, Hyderabad 500046
(India)
1
1 Introduction
Stochastic differential equations are a natural way of describing the interaction
of a system with a random reservoir. They were introduced by Langevin[1] to
help explain Einstein’s theory[2] of small particles immersed in a fluid, as ob-
served by the biologist Robert Brown. More rigorous mathematical treatments
were later introduced by Ito[3] and Stratonovich[4]. They have now diffused
into many different areas[5] of physics, chemistry and biology. In recent times,
similarmodelshavebeenutilizedinevermorediversefields,includingengineer-
ing, economics[6] , and even sociology. The essence of a stochastic differential
equation is that it isolates a system of interest from the backgroundof random
events that may influence the system. Implicit in this formulation is the idea
thatthereservoir,orsourceofrandomfluctuations,evolveswithoutreferenceto
the system of interest. This simplifies the study of otherwise complex coupled
phenomena.
As anexample, the calculationof correlationfunctions, andhence the spec-
tra,ofphysicalsystemsnearphasetransitionsisofconsiderabletheoreticalinter-
est. Thesehavedynamicalpropertiesthatareoftenconvenientlydescribedusing
stochastic differential equations. However, commonly used analytic techniques
like linearization, frequently become invalid near phase-transition points. At
the same time, while numerical simulation is possible, this is a time-consuming
computational procedure without resulting in a great deal of insight. Thus,
there is a need for techniques that give analytical results. Rather surprisingly,
therearefewsystematicproceduresfortreatingnonlinearstochasticdifferential
equations under conditions where linearization is invalid.
In this paper we consider the spectra of physicalsystems that are described
by stochastic differential equations near a critical point phase transition. Such
differential equations have a near universal applicability, both for equilibrium
and non-equilibrium phase transitions [5]. For instance, they arise in theoret-
ical treatments of single mode lasers [5], inhomogeneously and homogeneously
broadenedtwomodelasers[7,9],andopticalparametricamplifiersnearthresh-
old [10, 11]. A number of useful theoretical techniques [12, 17] are known for
theseproblems,someofwhichimproveuponthetheconventionalZwanzig-Mori
projectionoperatormethod[18,19]. However,thesetechniquesarecumbersome
for systems in higher phase space dimensions.
Instead, we propose a simple, direct calculation which is based on the
stochastic differential equation. The resulting integral expressions can be clas-
sified diagrammatically, in a way that allows a straightforward calculation of
essential combinatoric factors. The results give a power series in time which
can be extrapolated to long times with reasonable accuracy in many cases. We
analyze two possible extrapolation techniques, namely the exponential of a ra-
tional function, and a series of simple exponential terms. Either method gives
excellent results at or below the critical point.
Above the critical point, we find differences in accuracy, and this can be
2
related to the dominant eigenvalue distributions for different types of equation
and observable. Convergence is slowest when the spectrum has characteristic
time-scales which are an exponential function of an equation parameter, as in
the one-dimensional cubic stochastic process above threshold, which involves
diffusion over a barrier. The exponential series method is preferable for simple
types of spectra with only one or two dominant eigenvalues,which turns out to
be the case for the N-dimensional cubic stochastic process at large N-values.
The rational function method is best for complex spectra without much range
incharacteristictime-scales. Anexampleofthisistheintensity(x2)correlation
spectrum of the one-dimensional cubic process, which can be represented with
remarkable accuracy using low-order rational function extrapolation.
Moregenerally,weexpectthatthismethodcanbeappliedtoanystochastic
differential equation where conventional linearization methods are inapplicable
due to large nonlinear terms. Under these circumstances, it may be useful to
have a nonlinear solution of the type derived here, as a starting point for a
perturbative or asymptotic analysis. For these reasons, it is useful to analyze
the simple cubic nonlinear case in detail, both as a test case for the stochastic
diagram method, and as an elementary stochastic process of intrinsic interest.
2 Stochastic Equations
The method of stochastic diagrams to calculate solutions to stochastic differ-
ential equations is normally applied in the frequency domain, where it corre-
spondstoaperturbationtheoryexpansioninasmallcouplingconstant[20,21].
In these applications, there is a close resemblance to Feynman diagram tech-
niques. In both cases, the starting point of the iterative method is the ap-
proximatelinearizedsolutiontotheproblem,whichbecomesthesolutiontothe
entirecorrelationfunctioninthelimitasthecouplingconstantapproacheszero.
Frequency domain stochastic diagrams have many useful applications, includ-
ing the stochastic quantization approach to quantum field theory. An essential
difference between these methods and Feynman diagrams, is the appearance of
stochastic terms that are averagedover at a later stage.
We choosehereto applystochasticdiagramstothe time domaincorrelation
function. This has the advantage that there are no singularities in the series
expansioncoefficients,evenatacriticalpoint. Acorrespondingdisadvantageis
thatthelongtimecorrelationfunctionscannotbedirectlycalculated,andmust
be approximated by an extrapolation procedure that is based on some known
propertyofthe solution. Inthe examplesgivenhere,weuseeithersimpleexpo-
nentialsorlogarithmicrationalfunctionextrapolation,whichresultsinanalytic
expressionsfor the approximatecorrelationfunction. A directcomparisonwith
numerically calculated spectra will be used to demonstrate the great accuracy
of this procedure in calculating spectra near critical points. It is less accurate
abovethresholdinthe bistablecaseswherestochastic‘barrierhopping’or‘tun-
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neling’ can occur, resulting in widely differing eigenvalues. This results in a
reduced precisionfor the extrapolation. Methods based on multiple time scales
can be used in these cases.
Surprisingly, in the closely related higher-dimensional phase-diffusion prob-
lem,stochasticdiagramsgivegoodresults. Anexampleofthisisthelaserabove
threshold. This is because the long time-scale here is not exponentially long,
as it is in the tunneling cases. Good results are also found for non-bistable
variables like the intensity, even when the underlying equations are themselves
bistable. This is because the extrapolation is carried out in terms of the cor-
relation function, which has a different behavior to the underlying stochastic
variable. In order to illustrate these various cases, we start with a very general
form of stochastic differential equation.
The equations we wish to treat are of the form
x˙ =A(x)+B(x) ξ(t) , (1)
·
where the real noise sources ξ (t) have zero mean and are delta-correlated in
i
time so that
<ξ (t)>=0 ;<ξ (t)ξ (t′)>=δ δ(t t′) . (2)
i i j ij
−
Here x is a real vector of n-dimensions, A is an n-dimensional real polynomial
vector function of x and B is an n m dimensional real polynomial matrix
×
function of x. The vector ξ(t) is an m-dimensional real Gaussian stochastic
process, interpreted in the Itoˆ sense[22], for computational simplicity.
2.1 Iterative solutions
The method of stochastic diagrams consists of performing an iterative solution
for x(p)(t) so that
t
x(p)(t)=x + dt′[A(x(n−1)(t′))+B(x(n−1)(t′)) ξ(t′)] , (3)
0
Z ·
t0
where x(0)(t) x x(t ). Next, correlation functions of the typical form
0 0
≡ ≡
(t,t )=<x (t)x (t )> <x (t)><x (t )> , (4)
ij 0 i j 0 i j 0
G −
areevaluatedtopthorder,resultinginanexpansionof (t,t )asapowerseries
ij 0
G
in τ = t t for τ > 0. For any given term p in the power series, iterations
0
−
must be carried out until all possible terms in τp are evaluated. The result
of the iterations consists of integrals over time which will be represented as
directed lines in the stochastic diagrams. In addition, there are polynomials in
variables(representedasvertices),initialconditionsinthevariables(represented
as terminating arrows, and treated as delta functions at the initial time) and
stochastic terms (represented as crosses).
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2.2 Cubic stochastic process
Thus, for example, the solution of the well-known cubic stochastic process [22]
x˙ = x3+ξ(t) , (5)
−
has a first iteration, starting from a known initial value x(0)=v at t=0, of
t
x(1)(t)=v+w(t) dt′v3 . (6)
−Z
0
where w(t) = tdt′ξ(t′). More generally, the nth iteration in this simple one-
0
dimensional caRse is written
t
x(n)(t)=v+w(t) dt′(x(n−1)(t′))3 . (7)
−Z
0
Thus, we can expand the second term in the iteration as
t
x(2)(t) = v+w(t) dt′[v+w(t′) t′v3]3
−Z −
0
t
= v+w(t)+ dt′[(t′)3v9 3(t′)2v6w(t′) 3(t′)2v7 6t′v4w(t′)
Z − − −
0
+ 3t′v3w(t′)2+3t′v5 w(t′)3 3v2w(t′) 3vw(t′)2 v3] . (8)
− − − −
We seethat eventhis simple exampleleads to a largenumber ofdistinct terms,
whichneed to be classifiedin a systematic way. In particular,while the leading
term in the integral is of order t4, there are other terms of lower order present,
includingthestochastictermsandatermofordertwhichcomesfromtheinitial
condition.
2.3 Stochastic diagrams
The next term in the iteration involves a cubic integral of x(2)(t), and clearly
the combinatoricfactors involvedare more complex to three and higher orders.
In order to simplify the counting of these factors, a diagrammatic classification
can be introduced at this stage. In this classification, the terms are given di-
agrammatically to first order in Fig (1). To second order, all possible terms
in x(1)(t) appear as ‘legs’ on the nonlinear vertex, to the next higher order, as
shown in Fig (2).
Not all terms will contribute to the same order in a power series in τ, since
the expectation value ofa product oftwo stochastic integrals is proportionalto
τ, while the product of two deterministic integrals is proportional to τ2. This
means that a reordering of the sequence is needed, to obtain a series of terms
to a given order in τ. The rules are simple: all vertices counts as one order in
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τ while stochastic terms count as half an order in τ and initial values as zero
order. Ifthetermsinthereorderedseriesarelabeledasx˜(n)(t)wecanrepresent
them according to Fig (3). On expanding all the relevant terms in Fig (3), we
obtain:
x(0)(t) = v
x(1/2)(t) = w(t)
e
x(1)(t) = v3t
e −
x(3/2)(t) = 3v2w(t)
e −
x(2)(t) = 3[ vw2(t)+v5t2/2]
e e
−
t t
x(e5/2)(t) = w3(ft)+9v4 w(t′)dt′+6v4 w(t′)t′dt′
− Z Z
0 0
e f t e t
x(3)(t) = 3v3 w2(t′)t′dt′+9v3 w2(t′)dt′
Z Z
0 0
e t f 3
+ 18v3 w(t′)w(t′)dt′ t3v7 t3v7. (9)
Z − − 2
0
Here we have introduced the notation eof:
t
wn(t) wn(t′)dt′. (10)
≡Z
0
f
Further rules in stochastic calculus (of the Itoˆ variety)are that the expectation
values of the products of initial terms with stochastic terms decorrelate to all
ordersatlatertimesandalloddproductsofstochasticintegralsaveragetozero.
This means that the only surviving terms in the expectation value must be
ij
G
the terms of integer order in the series. For other types of expectation values
(involving polynomials in x(t)), these extra terms must be retained.
If we take expectationvalues ofthe relevantstochastic terms, they havethe
structure:
w2(t) = t
h i
w(t)w(t) = w2(t) =t2/2. (11)
h i h i
Hence, on decorrelating,eintegrating overftime, and combining all the relevant
terms up to third order we obtain (for the average and correlation function of
x(t)):
3 1
x(t) = v tv3+ t2(v5 v)+ t3(11v3 5v7)
h i h − 2 − 2 − i
3 1
x(t)x(0) = v2 tv4+ t2(v6 v2)+ t3(11v4 5v8) . (12)
h i h − 2 − 2 − i
These results are valid for an arbitrary initial distribution function. If carried
outtohigherorders,itisclearthattheycandescribeeitheratransientprocess,
or else a steady-state correlationin the time-domain, to any order in time.
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3 N-dimensional cubic stochastic process
Having introduced the stochastic diagram method, we now apply it to the N-
dimensional cubic stochastic process
x˙ (t)= η x f x x x +ξ (t) ; i,j,k,l=1, ,N . (13)
i i i ijkl j k l i
− − ···
Here summation over repeated indices is implied. The coefficient f of the
ijkl
cubic terms can be taken to be symmetric in the the last three indices without
any loss of generality. This stochastic equation, with appropriate choice of
parameters accommodates the stochastic equations that have been considered
in the context of single and two mode lasers and optical parametric amplifiers
[1-6]. The quantities of interest are the equilibrium correlationfunctions
(n)(τ)= lim [<xn(τ +t )xn(t )> <xn(τ +t )><xn(t )>] . (14)
Gij t0→∞ i 0 j 0 − i 0 j 0
This equation now has the added feature of a linear loss/gain term η. When
η > 0, there is additional damping, and the system is below threshold in the
usualsense. Thedeterministiccriticalpointisatη =0. Whenη <0,thesystem
has linear gain (like a laser above the lasing threshold), and the system is then
above the critical point. However,it is worth noting that as the dimensionality
increases, this type of classification which comes from a linearized analysis is
rather misleading. In fact, the enlarged phase-space volume means that noise
sourcesbecomeincreasinglyimportantatlargedimensionality-tothepointthat
there is a reduced distinction between the above and below threshold cases.
3.1 Steady-state behavior
Steady-state behavior is most readily analyzed if, for simplicity, we confine
ourselves to the following equation with N-dimensional rotational symmetry:
x˙(t)= ηx x(x x)/N +ξ(t) . (15)
− − ·
This corresponds to defining the cubic coefficient as: f = [δ δ +δ δ +
ijkl ij kl ik jl
δ δ ]/(3N) .
il jk
This stochastic equation has what is known as detailed balance - and hence
an exact solution in the steady-state, found by examining the corresponding
Fokker-Planck equation:
∂ ∂ 1 ∂2
P(t,x)=L P(t,x)= [η+x x/N]x + P(t,x) . (16)
∂t FP (cid:20)(cid:18)∂x · i 2∂x2(cid:21)
Xi i i
The equilibrium distribution is P (x) = exp[ V(x)], where V(x) is a
e
N −
potential function given by:
V(x)=ηx x+(x x)2/2N . (17)
· ·
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Thestochasticequationalsohasanexactrelationshipbetweenthemoments
of different orders in the steady-state, which can be easily derived from the
variable-change rules of Ito stochastic calculus. These are:
(n+1) =(2n+N 2) (n−1)/(2N) η (n) (18)
M − M − M
Herewehavedefined (n) = [x x]n /N,asaconvenientnormalizedform
e
M h · i
of the moment. Although these recursion relations are useful, the mean-square
fluctuationshavetobecalculatedfromthepotentialsolutionsgivenabove. The
quantity (1) = x x /N can therefore be computed explicitly and is given
e
M h · i
by
N N +1 N 1
(1) = U ,√Nη /U − ,√Nη , (19)
M r 4 (cid:18) 2 (cid:19) (cid:18) 2 (cid:19)
where U(a,x) denote the Whittaker functions [23]. For η = 0, this expression
simplifies to
2 N +2 N
(1) = Γ /Γ . (20)
M rN (cid:18) 4 (cid:19) (cid:18) 4 (cid:19)
3.2 N = 1 case
An important property of this potential in the one-dimensional case of N = 1,
is that it possesses a potential barrierat x=0, if η <0. This means that there
is a progression from a stable ‘below-threshold’ region for η > 0, (where x = 0
is the deterministic stable point), to a critical regionfor η =0 characterizedby
largefluctuations,andthento a bistable regionforη <0. This ischaracterized
by local stability in the two potential wells at x= η .
± | |
The N = 1 case has been well-studied in terms opf its eigenvalue spectrum.
Any one-dimensionalFokker-Planckequation can be transformed to an equiva-
lent Schroedinger equation problem with imaginary times[12], by introducing a
Schroedinger operator. In this case, it has the form:
L = exp[V(x)/2]L exp[ V(x)/2]
S FP
−
1 ∂2
= [2V′′(x) (V′(x))2]/8+
− 2∂x2
1 1 ∂2
= [x3+ηx]2+[η+3x2]/2+ . (21)
−2 2∂x2
At large positive values of η, the corresponding Schroedinger potential re-
ducestoaharmonicoscillatorproblem,withquadraticpotential. Transforming
back to real time, the eigenvalues of the Fokker-Planckoperator are of form:
L P (x) = λ P (x) , (22)
FP n n n
−
where λ = nη. Physically this is easy to understand. The equation is domi-
n
nated by the linear decay rate η, and integer multiples of η will occur through
the decay of integer powers of the variable x.
8
At large negative values of η, the equation is bistable, and there are two
principle eigenvalues. A fast equilibration inside each potential well occurs,
with aneigenvalueof λ =2η in the limit oflargeη. A slow decayalsooccurs
f
| |
through diffusion over the barrier. Ignoring terms in η of order (1) in the
| |
pre-factor, this gives a slow eigenvalue:
ln[λ ] ∆V +ln[η ] η4/2+ln[η ]. (23)
s
≃− | | ≃− | |
It is significant for the stochastic diagram method, that this eigenvalue is
exponentially small in the limit of largeη. Thus, we cannot expect an accurate
estimate of the eigenvalue with any technique involving a finite series of terms
in η, andanycorrespondingspectrumin whichλ is significantwill notbe able
s
to be estimated with a finite expansion in powers of η.
3.3 Large N case
For N > 1, a similar progression from below to above the critical point holds
deterministically, except that there is no bistable region. Instead, for η << 0,
there is a region characterized by neutral stability in the subset of phase-space
where x η N. Thus, there is a continuum of possible deterministically
| | ≃ | |
stable behavipour. This phenomenonissometimes calledspontaneoussymmetry
breaking. To show this more clearly, consider the distribution P (R), in the
R
variable R = x2/N. This has a steady-state potential of V (R) = N[ηR+
R
| |
R2/2 (1/2 1/N)ln(R)]. As a result, for increasing N, the distribution in
− −
R is peaked more and more strongly near the value R = ( 2+η2 η)/2. In
η
−
fact, due to the increase in phase-space volume as R increapses, there is always
an outward ‘entropic’ force even when η > 0. This means that the stochastic
equation at large N is not described well by the deterministic stability theory.
Inthislimit,theradiusapproachesafixedvalue,duetothebalancebetween
theoutwardentropicforceduetoincreasingphase-spacevolume,andtheinward
force due to the nonlinearity. Thus, the moments (n) all factorise, and are
M
given by:
(n) =([ 2+η2 η]/2)n . (24)
M −
p
The recursion relation for moments now simplifies, and it is straightforward to
verify that the above solution does satisfy the recursion relation.
Generally, in a stochastic equation, spontaneous symmetry breaking is ac-
companied by a type of phase-diffusion, or tangential diffusion in a surface of
dimension N 1. Hence, the lack of bistability for any N greater than one
−
results in a dynamical behaviour in which diffusion still occurs, but with a re-
duced dimensionality. These two types of above-threshold behaviour result in
different dynamical regimes for the resulting correlation functions and spectra.
In both cases, the above threshold dynamics typically involves more than dif-
ferenttime-scale. Theradialrelaxationtoastable pointwithinapotentialwell
9
inthe R-spaceequationsgenerallyoccursmuchfasterabovethresholdthanthe
tangential diffusion.
There are corresponding changes in the large-N dynamics, and the physical
explanation for this is interesting. In the limit of N , the fast radial
→ ∞
equilibration takes place in an approximately quadratic potential well at all
values of η. It may be noted that the corresponding (Itˆo) stochastic equation
for radial equilibration involves noise in a multiplicative way. In the general
case, we find that:
R
R˙ = 2ηR 2R2+1+2 ξ(t) , (25)
− − rN
where ξ(t)ξ(t′) =δ(t t′). Aswellashavingmultiplicativenoise,thisequation
h i −
also shows why the stochastic equation trajectories are confined to an increas-
ingly small region in R-space, as N increases. This occurs because the relative
size of the noise term in the radial equation decreases as N increases. Thus,
radial equilibration takes place with a fast relaxation rate of
λ =4R 2η=2 2+η2 . (26)
f η
−
p
Inthetangentialdirection,diffusiontakesplaceonahyper-sphericalsurface
of fixed radius defined by
x = NR . (27)
η
| |
p
Suppose we define a coordinate system so the diffusion starts with a radial
coordinateofx = NR attimet=0. Forsmalltimes,theother(tangential)
1 η
coordinates obey thpe diffusion equation, so that
x2 =t , (28)
h ji
for j > 1. Since the radius is fixed by the radial equation, it follows that
this corresponds to angular diffusion. Projecting each angular variable in turn
onto a radius vector in a lower dimensional subspace reduces the length of
the resulting vector. Finally, in the subspace of one dimension containing the
original (starting) vector, we have:
x (t) NR [1 t/NR ](N−1) NR exp[ (1 1/N)t/2R ] . (29)
1 η η η η
h i≃q − ≃ − −
p
This corresponds to a much slower tangential relaxation rate of
λ =(1 1/N)/(2R ) (30)
s η
−
in the large N limit. We note that this is not exponential in η, unlike the one-
dimensional case. Similar behaviour would occur in the case of finite N and
large,negative η, which is also dominated by the tangentialdiffusion caused by
spontaneous symmetry breaking. However, for the case of finite N values, the
slow eigenvalue must reduce to λ =η in the limit of large enough positive η.
s
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