Dynamics of Hierarchically Clustered Cooperating Agents
eivind.tostesen.dk/theses/tostesen_master_thesis.pdf
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A metastable state is a con guration for which every spin ip increases the energy
Alocalminimumofthecurvesmeansthatoptimaltransitionsin crease the energy But optimal transitions have the lowest energy increases so a local minimum corresponds to a metastable state
The parabolic hill
Self similargeometry
The graphs of En haveaverysimpleself similarstructure Theyaremadeup of parabolic #hills$ each in turn made up ofz times as many parabolic hills on a smallerscale etc etc There is a factor of z di erencein heights of the hills between succesive levels This can be seen from eq
Choose a level m In g
we examine the energy of the states nj uzm jzm where u is an arbtrary integer and j
z We need to know the digits cnj l for these states in order to use eq
For l mthese digits are zero For j wehave nj xx j z where xx z u zm Fromeq
weget Enj En hn m zmj j z j z mJ
For j wecannothave cnj m j Instead we have to carry to the next digits More than one digit may be a ected this way but eq
still holds This may be seen in the following way hn m is the distant eld from spins outside the block of sizezm that is being lled for n n n During this lling hn m is therefore unchanged and equal to hn m The change in energy between n and n is only determined by hn m because energy in internalinteractionsbetweenthespinsintheblockisconservedwhenthey are ipped simultaneously Only interactions with the outside give a change so En En hn m zm
with a minusbecause the spins are before they are ipped This shows that eq
also holds for j From eq
it is seen that hn m is the slope of the line in g
connecting points and The term j z j z mJ givesaparabola see g
Chapter
The free energy landscape
In this and the following chapter we only consider nite systems
Relaxation
Wewanttostudythedi usionofthesysteminitsenergylandscapewith A as the starting point at time t We saw in the previous chapter that we havearuggedenergylandscapewithmanymetastablestates Accordingto the standard picture of relaxation and broken ergodicity in such systems the system will explore bigger and bigger parts of the free energy landscape in the following way On a given timescale the system will be con ned to a component whichisthepartofstatespaceaccessiblefromthestartingpoint without having to surmount a free energy barrier higher than F where eF T and T is the temperature Being con ned means that there is a small probabilityofescapingthecomponentonthattimescale Onthistimescale the system is ergodic within the component i e it revisits all parts of the componentwithfrequenciesapproximatelygivenbytheGibbsdistribution restricted to the component On smaller timescales the system will only haveequilibratedwithinsmallercomponents Weimaginecomponentsas lakesinthefreeenergylandscape withthewaterlevelequaltothesmallest free energy barrier to escape As time increases the water level increases logarithmically and lakes grow into bigger lakes Disjoint lakes in di erent parts of the landscape merge in a hierarchical way with increasing water level
This picture is being to explain dynamics of disordered systems e g age ing in spin glasses but the intuition is based on di usion in a one dimensional state space For some systems a pond and outlet picture applies instead where ponds visited earlier are not revisited
whichisaddedtothelinearterm hn m zmj This is the hill and the factor z m givesitsscaling
[PDF]
Metastable States in Spin Glasses and Disordered ...
arxiv.org/pdf/cond-mat/9908455
arXiv
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by CM Newman - 1999 - Cited by 25 - Related articles
dynamical process with flips of lattice animals up to size M and starting from a deep .... volume spin configuration whose energy cannot be lowered by the flip of any finite subset .... of the metastable states falls to zero as volume increases (
[PDF]
Monte Carlo investigation of the Ising model
https://www.physics.ohio-state.edu/~braaten/statphys/Ising_MatLab.pdf
by T Fricke - 2006 - Cited by 5 - Related articles
This we do as follows: We place the N particles in any configuration, for example, in a regular ... These are the states with all spins aligned, either all up (M = 1) ... At low temperatures there appears to be also a class of 'metastable' states in which approximately half ... will increase the energy slightly from the ground state
[PS]
Shapes of growing droplets | a model of escape from a ...
ftp://ace1.ma.utexas.edu/pub/mp_arc/mp_arc/html/.../93-202.ps.gz
by R Koteck - Cited by 34 - Related articles
ABSTRACT: Nucleation from a metastable state is studied for an Ising ..... Lemma 2.1 Let be a con guration whose energy increases under every spin ip. .... Then every spin ip leads to a change of energy by a minimal amount of at least h.
[PS]
93-202.ps.gz - Department of Mathematics
https://www.ma.utexas.edu/.../93-202.ps.gz
University of Texas at Austin
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by R Koteck - Cited by 34 - Related articles
ABSTRACT: Nucleation from a metastable state is studied for an Ising ..... Lemma 2.1 Let be a con guration whose energy increases under every spin ip. .... Then every spin ip leads to a change of energy by a minimal amount of at least h.Metastability - Wikipedia, the free encyclopedia
en.wikipedia.org/wiki/Metastability
Wikipedia
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Metastable states of a ferromagnet on random thin graphs_ ...
wenku.baidu.com/view/b8542d76a417866fb84a8e79.html
[PDF]
Here
www.icn.ucl.ac.uk/.../BurgessMoore89.pdf
University College London
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by N Burgess - 1989 - Cited by 5 - Related articles
Mar 1, 1989 - Thus the solution of minimum cost lies in a vast configuration space containing many local minima whose numbers often increase exponentially with N. ... energy of the system we will refer to metastable states rather than locally optimal solutions. ... We found that in these states the mean energy per spin E =.
[PS]
Metastable States in High Order Short-Range Spin Glasses
https://www.bioinf.uni-leipzig.de/.../99-09-062.ps.g...
University of Leipzig
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by VM de Oliveiraa - Cited by 14 - Related articles
where A is the adjacency matrix of the configuration space (V, X), D is the diagonal ... fact that all Ising spin models. H (x) = ∑. (i1<i2<...<ip). Ji1i2...ip xi1 xi2 ...xip ... like exp(αN) with α increasing from 0.199 for p = 2 to ln2 ≈ 0.692 for p ... considers specific classes of metastable states (e.g. those possessing a given energy.Metastable States of the Fuzzy Spin Mo~el. I - Progress of ...
ptp.oxfordjournals.org/content/84/2/213.full.pdf
by T Kawasaki - 1990 - Cited by 5 - Related articles
configuration of the spins is considerably ramified and stability of the .... spin magnitudes can have any values randomly and continuously between 0 and S site ... stand for up- and down-spi~s, re- ... the width goes to zero with increasing sample ... frequency distributions of the metastable energy states are plotted in Fig.
[PDF]
(Here, S is the entropy and {S} is a spin configuration. ... Chapter 2 u(ri − rj) = u(r) is the interaction energy and U the potential of the box walls. ... tion of state in a mean-field approximation, we neglected all the correlations be- tween the molecules. ... The ultimate stability limit of the metastable state is called the spinodal line.
Chapter 2 Classical Models
www-f1.ijs.si/~vilfan/SM/ln2.pdf
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