All Questions
Tagged with complex-systems statistical-mechanics
48 questions
9
votes
2
answers
324
views
What is the reason for power laws appearing at phase transitions?
Let's consider a system with order parameter $\rho$ (e.g. density in liquid-gas transition, magnetization in Ising model) and control parameter $\tau$ (usually the normalized temperature, but it could ...
- 294
1
vote
0
answers
73
views
Given an infinite amount of time, will every possible combination of matter pop into existence?
Apparently it is true that when the universe is in the state of heat death, quantum fluctuations will eventually produce every combination of matter, no matter how unlikely, given an infinite amount ...
- 137
1
vote
2
answers
128
views
Is it possible to define a universal formula for chaos?
I've been working on a chaos project. I have noticed that there are several formulas to find the behavior of chaos, for example:
The logistic map is a simple equation that exhibits chaotic behavior ...
- 37
-2
votes
1
answer
113
views
Advanced math courses for theoretical physics students [closed]
Theoretical physics studies concerning statistical mechanics, dynamical systems and analytical classical mechanics all require working knowledge of mathematical concepts and theories (e.g. manifolds, ...
0
votes
0
answers
25
views
Average resultant length $L$ of an active string initially of length $L_\mathrm{0}$ after a single point mutation
I stumbled upon this research article, where they define a digital organism as an abstract minimal model of an evolving predator-prey system as follows:
An organism is defined via its genome of fixed ...
2
votes
1
answer
64
views
Meaning of Gibbs average
This might sound a very amateur question but I couldn't find the answer anywhere. Simply, what is the meaning of Gibbs average? I've came across this term on the paper: Transmission of Information ...
- 61
2
votes
1
answer
389
views
Binder cumulant method for non-Gaussian distributions
In the Ising model, we know that the order parameter $m$ has a Gaussian distribution for temperatures below the critical point. Measuring the exact point where this phase transition takes place was ...
0
votes
0
answers
52
views
Are there any resources for simulating the Kuramoto model/solving the system of coupled ODEs for the oscillators?
I want to simulate the Kuramoto model and I am following a review by Strogatz to read up the theory. Are there resources that explain how to simulate the model and solve the coupled ODEs, or the mean ...
3
votes
1
answer
79
views
Stability of Chemical Reactions [closed]
Given the following reactions:
$$A + X \xrightarrow{k_{1}} 2 X$$
$$Y + X \xrightarrow{k_{2}} 2 Y$$
$$Y \xrightarrow{k_{3}} B$$
I was able to write the following rate equations for the concentrations:
$...
- 653
2
votes
2
answers
740
views
Understanding ergodicity and what an ergodic system is
I am trying to understand the concept of ergodicity/ergodic system in physics, but because my understanding of phase space, its elements is a bit unclear,I have trouble understanding the former. ...
- 1,624
9
votes
2
answers
459
views
How has Parisi's nobel-prize winning work been applied to all kinds of complex systems?
As discussed briefly in this APS Physics editorial, Nobel Prize: Complexity, from Atoms to Atmospheres, the most important works of the recent Nobel prize winning physicist concern the study of the ...
- 1,678
3
votes
1
answer
170
views
Complicated partition function emerges from simple constraints
I'm working on a statistical model which involves many degrees of freedom $𝑖=1...𝑆$. Each degree of freedom is described by a gamma distribution with its own parameters, which we will assume to be ...
- 31
1
vote
0
answers
105
views
A quick introduction to Complex Ginzburg-Landau Equation
The complex Ginzburg-Landau equation is given as follows:
$$
\partial_{t} A=A+(1+i b) \Delta A-(1+i c)|A|^{2} A.
$$
For some reasons, I need to quickly understand what the parameters in the equation ...
2
votes
2
answers
194
views
When is molecular chaos dynamical chaos?
It is very common to have uncorrelated velocities in chaotic dynamical systems. Yet, we should be wary in equating the two quite different meanings of chaos.
Instead of matching dynamical chaos to ...
- 202
1
vote
1
answer
531
views
Fokker--Planck equation - naming a vector field
A Fokker Planck equation for the prob. density $\rho$ may be written in the form of a continuity equation
$$\frac{\partial \rho(x,t)}{\partial t} = - \nabla \cdot \left[ g(x,t) \rho(x,t) \right].$$
...
1
vote
1
answer
44
views
Summing over Adjacency Matrices in Partition Function
I'm currently reading a paper (abstract here) on the statistical mechanics of Random Geometric Graphs, and they start with the statistical mechanics of hidden variable graphs.
They've taken $a_{ij}$ ...
2
votes
0
answers
48
views
Completely Integrable Frustrated Lattice Systems
The Toda lattice is a prime example of a lattice system that is completely integrable, in the sense that it admits a Lax pair,
https://doi.org/10.1143/PTP.51.703,
making it easy to find soliton ...
- 713
4
votes
1
answer
265
views
A general way to combine equilibrium constants in reaction networks?
I have a network of states, each linked with neighboring states by unique forward and reverse transition rates ($k_{f}$ and $k_{r}$) - let's just say these are chemical species with multiple ...
- 95
37
votes
3
answers
5k
views
Why can't many models be solved exactly?
I have been told that few models in statistical mechanics can be solved exactly. In general, is this because the solutions are too difficult to obtain, or is our mathematics not sufficiently advanced ...
- 503
0
votes
1
answer
69
views
Perturbing PDF with spatial dependent perturbation
Let's consider a PDF $\rho(x)$, with normalization 1. Let's perturb it in the following way:
$$
\rho(x+\varepsilon F(x) ),
$$
with $\varepsilon$ small. I impose that the perturbed PDF is again a PDF ...
4
votes
2
answers
477
views
Kullback-Leibler divergence as a measure of irreversibilty?
I watched this recent KITP webinar on Nonequilibrium thermodynamics for active matter yesterday. I saw that KLD(Kullback-Leibler divergence) is used as a measure to quantify irreversibility in the ...
- 416
3
votes
1
answer
131
views
Critical exponent relation for neural avalanche dynamics
I am trying to understand the origin of equation (4) in this paper. Per the arguments of Touboul and Destexhe, a power law distribution for avalanche size and duration, as well as observed data ...
- 235
2
votes
0
answers
56
views
"Unnatural" Hamiltonian systems from a statistician's perspective
I would like to learn more about "unnatural" Hamiltonian systems, that is, systems whose energies cannot be written as
$$H(p,q) = K(q) + U(p).$$
I have seen the term "natural" applied to systems ...
- 121
2
votes
0
answers
465
views
Liouville theorem and the ergodic assumption
I am following a course on statistical mechanics. My instructor presented us the following Liouville theorem in two (claimed) equivalent ways:
Differential statement: The probability distribution $\...
5
votes
0
answers
695
views
Textbooks about spin-glasses for beginners
I am a Ph.D. student in Physics attending my second year. I would like to ask you whether you know any good textbook about spin-glasses (and physics of complex systems in general) for beginners. ...
7
votes
2
answers
1k
views
Statistical Mechanics & Dynamical Systems
As a (theoretical) physics student I've taken (advanced) undergrad courses in both statistical mechanics and dynamical systems (which was purely mathematical, treatment of nonlinear differential ...
2
votes
2
answers
1k
views
Ott-Antonsen-Ansatz
I'm reading a paper https://doi.org/10.1063/1.2930766 about the Ott-Antonsen-Ansatz that is used to describe the dynamics of global coupled oscillator. There is a computational step from equation (4),(...
- 51
2
votes
1
answer
688
views
Can Chaos Theory be used to explain the Ising model in paramagnetic phase?
Is it possible? How can I explain the randomness of spins in the paramagnetic phase with chaos theory? In this case, is the randomness apparent?
If yes, I think the temperature would be a reasonable ...
- 389
0
votes
0
answers
32
views
Nutrient turning into bacteria according to the Slow Growth Law instead of the 2nd Law
I'm reading a paper by Charles H. Bennett on different metrics for complexity in the physical world: http://cqi.inf.usi.ch/qic/94_Bennett.pdf
In a paragraph on page 35, where he describes ...
- 111
1
vote
1
answer
1k
views
What are spatially extended systems?
i am reading about coupled map lattice and stuck on what exactly is the meaning of spatially extended systems. Please help.
7
votes
1
answer
624
views
necessary and sufficient conditions for an isolated dynamical system which can approach thermal equilibrium automatically
Given an isolated $N$-particle system with only two body interaction, that is
$$H=\sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}+\sum_{i<j}V(\mathbf{r}_i-\mathbf{r}_j)$$
In the thermodynamic limit, that ...
- 6,071
4
votes
1
answer
345
views
How to compute entropy of networks? (Boltzmann microstates and Shannon entropy) [closed]
I also asked in SO here a few days ago, thought it may be also interesting for physics-related answers.
I would like to model a network as a system.
A particular topology (configuration of edges ...
- 143
4
votes
3
answers
940
views
What is the definition of "Complexity" in physics? Is it quantifiable?
I don't know much about the discipline of "Complex systems studies" but I know in the field of "Statistical mechanics" there is much talk about the "Complexity of the system&...
- 872
2
votes
2
answers
156
views
How do we understand the dynamics and nonequilibrium?
In our physics classes,we have learned lots of laws to describe the motion of particles,such as Newton's second Law
$$F = m \ddot x$$
in classical mechanics,and also the famous Schrödinger ...
- 1,785
1
vote
0
answers
129
views
Which condition is stronger - ergodicity or mixing?
Reading a statistical physics book, I've encountered the following assertion (without further explanations):
[..] the presence of dynamical instability makes the trajectory of a
system much more ...
- 647
3
votes
2
answers
934
views
Is there a relation between complexity of a system and entropy?
Disclamer: I'm not a physics professional, so pardon me if the question is stupid/incomperhensible/generally doesn't make sense. And I've googled it, but didn't find an answer.
Getting to the point, ...
- 131
7
votes
4
answers
1k
views
Physical distinction between mixing and ergodicity
How can one in a very contrasting manner distinguish between the physical meaning of mixing dynamics and that of ergodic dynamics? More precisely, is one a stronger condition than the other? (which ...
- 4,800
2
votes
2
answers
467
views
Can an Ergodic dynamical system approach equilibrium?
An ergodic dynamical system $(\Omega,\phi^t,\mu)$ is such that the time average $\bar{f}$ of every function $f\in L_1(\Omega,\mu)$ equal the space average $\langle f \rangle_\mu$, i.e. the system ...
- 449
3
votes
2
answers
306
views
Resources on Master Equations
Presently I am reading about "Introduction to dynamical process theory and simulation" which uses the notion of Master Equations to solve Markov process.
I am very new to this. Can someone provide me ...
4
votes
2
answers
480
views
Reference request for exactly solved models in statistical mechanics
Can someone recommend a textbook or review article that covers exactly solved models in statistical mechanics, such as the six- or eight-vertex models? If there is literature at the undergraduate ...
20
votes
5
answers
10k
views
What are some of the best books on complex systems and emergence?
I'm rather interested in getting my feet wet at the interface of complex systems and emergence. Can anybody give me references to some good books on these topics? I'm looking for very introductory ...
8
votes
1
answer
335
views
Deviation from power law distribution of earthquakes
One of the most accepted frameworks for the relationship between the magnitude and frequency of an earthquake is that of the critical phenomena. In this framework, the magnitude of events must be ...
- 836
2
votes
1
answer
174
views
Lacking of scale and distribution moments
Given a physical random variable x, $E(x)$ and $E((x-<x>)^2)$ defines mean and variance. From a statistical point of view variance represents the statistic error (isn't it?). If variance is not ...
- 836
2
votes
1
answer
144
views
Are rainfalls critical phenomena?
By definition, rainfalls are transitions from vapor state to liquid state of water. Can I say that "by definition" rainfalls must viewed as critical phenomenon?
- 836
2
votes
3
answers
4k
views
The meaning of scale invariance in power law distribution
A function $f(ax)$ that satisfies
$$
f(ax)=a^\Delta f(x)\,\,\, (\Delta \in R)
$$
is said to be scale invariant. The most general function $f(x)$ that satisfies the previous condition is of the form
$$...
- 836
4
votes
1
answer
362
views
Scale invariance in sandpile model and forest fire model
I asked a similar question but the wrong way here. Because my intention was to ask about non thermodynamic system, I will be more specific:
What is the relation between critical behaviour and the ...
- 836
9
votes
1
answer
926
views
Scale invariance and self organized criticality
On wikipedia I have found this statement:
In physics, self-organized criticality (SOC) is a property of (classes of) dynamical systems which have a critical point as an attractor. Their macroscopic ...
- 836
15
votes
2
answers
4k
views
Which areas in physics overlap with those of social network theory for the analysis of the graphs?
I am studying social networks in terms of graph theory and linear algebra. I know that physicists have published and worked a lot in this field. This causes me to assume that there are sub-fields in ...
- 782