SISTEMAS COMPLEJOS
TEMA II – LA FÍSICA DE LOS SISTEMAS COMPLEJOS
INDICE
ENSEMBLES, MICROESTADOS, MACROESTADOS
FUNCIONES TERMODINAMICAS
TRANSICIONES DE FASE Y UNIVERSALIDAD
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INDICE
ENSEMBLES, MICROESTADOS, MACROESTADOS
FUNCIONES TERMODINAMICAS
TRANSICIONES DE FASE Y UNIVERSALIDAD
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STATISTICAL MECHANICS in a nutshell
Why stat.mech.?
Description of gases with Newton’s laws is impracticable!
How stat. mech.?
Statistical properties of ‘molecule collectivities’
What stat.mech.?
- Theory that connects microscopic (individual) dynamics to macroscopic (statistical)
collective properties:
Position, speed of particles (mechanics) pressure, temperature, volume, entropy of
gas (thermodynamics)
Relation to dynamical systems:
Ergodicity, molecular chaos, KAM theorem and stat mechs paradox
,
FUNDAMENTAL CONCEPT IN STAT. MECH.: ensemble!
Microstate:
A specific set of positions and velocities of N elements
Macrostate:
A macroscopic variable of a microstate (e.g. temperature, volume)
Each macrostate has many possible microstates (degeneracy).
Ensemble: set of different accessible microstates that compose a system.
- Microcanonical ensemble (fixed N,V,E) –microstates with same energy
- Canonical ensemble (fixed N,V,T) –microstates with same temperature
- Grand canonical ensemble
EXAMPLE 1
System: N2 leds in a lattice, each led is either YELLOW or BLACK
Macrostate: 30% YELLOW, 70% BLACK
Microstate A
Microstate B
Microstates A and B belong to the same macrostate, but they are different configurations!
The set of all possible configurations (microstates) with same macrostate is called an
ensemble
Question: which is the most probable microstate?
which is the most probable spatial distribution of yellow leds? … ENTROPY …
EXAMPLE 2
SYSTEM: pool balls.
Macrostate: the sum of velocities is zero.
Microstates?
Many possible microstates!
A. All balls have zero speed
B. All balls are placed in a circle and converge to its center with the same speed
C. Balls have random speeds (the sum of velocities converges to zero in the statistical sense)
Which is more probable?
Is C a microstate?
HOW DO WE DIFFERENCIATE / WEIGHT ALL THESE MICROSTATES?  INTERNAL ENERGY
INDICE
ENSEMBLES, MICROESTADOS, MACROESTADOS
FUNCIONES TERMODINAMICAS
TRANSICIONES DE FASE Y UNIVERSALIDAD
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FUNDAMENTAL QUANTITY IN STAT. MECH.: partition function!
Boltzmann factor
Suppose a system with N elements at temperature T
The probability that the system is in one of the microstates with energy Ei goes like
Partition function
Number of different microstates at a given temperature (each with different energy)
Bridges the gap from micro (individual) to macro (statistical)
The PROBABILITY of finding the system at temperature T in a state with energy Ei is
Partition function
FROM THE PARTITION FUNCTION WE CAN CALCULATE ALL THERMODYNAMICAL QUANTITIES
- Total energy
- Free energy
F
F
- Entropy
Microscopic
Macroscopic
Hamiltonian H  Partition function Z  Free energy F
Entropy S
Pressure P
Magnetization M …
Quantifies interaction strength
between elements
Partition function
FROM THE PARTITION FUNCTION WE CAN CALCULATE ALL THERMODYNAMICAL QUANTITIES
- Total energy
Entropy, disorder, information
- Free energy
F
F
- Entropy
- Temperature
Microscopic
Macroscopic
Hamiltonian H  Partition function Z  Free energy F
Entropy S
Pressure P
Magnetization M …
Quantifies interaction strength
between elements
Partition function
FROM THE PARTITION FUNCTION WE CAN CALCULATE ALL THERMODYNAMICAL QUANTITIES
- Total energy
- Free energy
F
F
- Entropy
- Temperature
Microscopic
Macroscopic
The inverse of the temperature is the cost
Hamiltonian
 Partition
function
Z
Free
energy F
of buying Henergy
from the
rest of
the
world.
Entropy is the currency being paid. Entropy
Inverse S
temperature is the cost in entropy toPressure
buy a P
Magnetization M …
unit of energy.
Quantifies interaction strength
between elements
Partition function
FROM THE PARTITION FUNCTION WE CAN CALCULATE ALL THERMODYNAMICAL QUANTITIES
- Total energy
- Free energy
F
F
- Entropy
- Temperature
Microscopic
Macroscopic
Hamiltonian H  Partition function Z  Free energy F
Entropy S
Pressure P
Magnetization M …
Quantifies interaction strength
between elements
Microscopic
Macroscopic
Hamiltonian H  Partition function Z  Free energy F
Entropy S
Pressure P
Magnetization M …
 Normalmente desconocemos el Hamiltoniano del sistema (ecuaciones de movimiento), o dicho
Hamiltoniano no puede obtenerse a través de una aproximación mecánica tipo minimización de
la acción (sistema no físico)
 La aproximación puede empezar un poco más ‘mesoescala’:
* ¿cuáles son los grados de libertad ‘interesantes’?
* ¿qué reglas de interacción entre las ‘partículas’ de mi sistema?
Estas reglas de interacción generan un ‘Hamiltoniano efectivo’
 Simulamos con un ordenador la ‘dinámica molecular’ (interacciones locales)
 Definimos y medimos variables globales (el análogo a los macroestados) que caracterizan los
fenómenos colectivos, emergentes, etcétera
Es decir:
este formalismo sirve para modelizar fenómenos colectivos dentro y fuera de la física
Microscopic
Macroscopic
Hamiltonian H  Partition function Z  Free energy F
Entropy S
Pressure P
Magnetization M …
“The molecules are like individuals, … and the properties
of gases only remain unaltered, because the number of
these molecules, which on the average have a given
state, is constant,”
“This opens a broad perspective, if we do not only think of
mechanical objects. Let’s consider to apply this method to
the statistics of living beings, society, sociology and so
forth.”
Ludwig Edward Boltzmann 1844 - 1906)
INDICE
ENSEMBLES, MICROESTADOS, MACROESTADOS
FUNCIONES TERMODINAMICAS
TRANSICIONES DE FASE Y UNIVERSALIDAD
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WHAT IS A PHASE TRANSITION?
OPTIMIZATION PROCESS:
MINIMIZATION OF FREE ENERGY F=E-TS
Phase: macroscopical state of a system.
- The stable phase will be the one that minimizes the free energy F of the system
- The stable phase is the one that
* minimizes the internal energy of the system
* maximizes his entropy
-Varying a parameter of the system (control parameter), the stability of the
macroscopic phases can switch : phase transition
WHAT CAUSES THE CHANGE OF STABILITY?
Symmetry breaking (one phase has more symmetry than the other)
 order parameter describes this symmetry breaking
CLASSIFICATION OF PHASE TRANSITIONS
Order parameter vs Control parameter
Order of the Derivative of the order parameter which is discontinuous (first
order, second order, etc) provides the transition ‘order’
EXAMPLE (first order transition)
Ice-water transition: Order parameter is discontinuous in the transition point.
-
-
Ice and water at 0ºC
Latent heat is needed not to
increase the temperature of the
solid but to change its symmetrires
Order parameter changes
discontinuously at the transition
EXAMPLE (second order transition)
Ferromagnetic-Paramagnetic transition
-
-
No latent heat
involved
Order parameter
changes
continuously at
the transition
Other
magnitudes
diverge
Second order/continuous/critical
The order parameter:
-is continuous,
- vanishes in the ‘disordered phase’ and is non-zero in the
‘ordered phase’
- Nonanalitic at transition (fluctuations diverge)
- Transition point is a critical point: fluctuations around this
point can have arbitrary sizes (scaling, power laws, scale
invariance).
- Perturbations close to the transition propagate at every
scale (long-range correlations). P(response) is a power law
- Universal behavior close to the transition point: scaling
laws, critical exponents, universality classes
-critical point is unstable, only reached by tuning a control
parameter such as temperature
Comment: Can systems self-organize around a critical
value?
,
PHASE TRANSITIONS VERSUS LOCAL BIFURCATIONS (valid analogy in mean field)
first order phase transition
vs saddle bifurcation
dy/dt = a + by2
second order phase transitionpitchfork bifurcation.
dy/dt = y – by3
These phenomena extend well beyond solid state physics
Phase transitions are the result of an optimization process
In a multicomponent system one can define
 Local rules (similar to Newton laws of motion)
 Develop a statistical mechanics approach
 Find different phases and in some cases
 Phase transitions
EXAMPLE I: ISING MODEL (MAGNETISM)
SOME MATERIALS BELOW A CERTAIN TEMPERATURE EVIDENCE A
SPONTANEOUS MAGNETIZATION
2 BASIC (OPPOSED) MECHANISMS:
-Tendency to allign with the local magnetic field to minimize energy
-Thermal noise induces disorder as a function of temperature T
Phase transition.
Global magnetization
1. T=0
well ordered
2. 0<T<Tc
ordered
3. T>Tc
disordered
PLD’s
ISING MODEL (numerical simulations – Monte Carlo)
T<Tc
T=Tc
T>Tc
EXAMPLE II: VOTER MODEL (SOCIOLOGY)
Voter model
spins are people opinion(red, blue)
EXAMPLE III: FLOCKING MODELS (COLLECTIVE PHENOMENA)
Noise-induced transitions: animal flocking
Flock1
Flock2
A simple model: Follow your neighbors !

v

i (t )
v j (t  1)  v0 
vi (t )
R
  j (t )
R
• absolute value of the velocity is equal to v0
• new direction is an average of the directions of
neighbors
• plus some perturbation ηj(t)
EXAMPLE IV: BONABEAU MODEL (SOCIOLOGY)
Social hierarchy generation: Bonabeau model with 2 opposite mechanisms
(1) Competición con retroalimentación: se
elige un agente i al azar y se mueve de modo
aleatorio a una de sus cuatro casillas vecinas. En
el caso de que la casilla esté vacía, el agente pasa
a ocuparla. Si está ocupada por un agente j se
produce un enfrentamiento. El agente atacante i
vencerá al agente j con probabilidad:
Donde  > 0 es un parámetro constante del sistema.
Si i gana, intercambia la casilla con j.
Si pierde, se mantienen las posiciones.
Después de cada enfrentamiento los status hi(t) y hj(t) se
actualizan sumando a su valor 1 en el caso del vencedor y
disminuyendo en 1 en el caso del derrotado.
(2) Relajación: se define un paso de tiempo en el sistema después de N
repeticiones de la
operación anterior (haya o no enfrentamiento).
Después de cada paso de tiempo todos los agentes multiplican su valor hi(t)
por un factor de
relajación (1 - ), donde 0 <  < 1.
MEDIDA DE LA JERARQUÍA
Una medida natural de la jerarquización o
diversidad de status del sistema es la desviación
típica de la distribución de las probabilidades
estacionarias:
Está acotada inferiormente con valor 0 y superiormente con valor 1. Actúa como
parámetro de orden del sistema.
Imagen de transición Bonabeau
Simulación numérica
A bajas densidades
la dispersión es nula:
hay igualdad entre
agentes.
A altas se genera
jerarquía bruscamente a
través de una transición de
fase
Generación de jerarquía en el modelo numérico de Bonabeau
• Sociedades pequeñas (manadas de animales, tribus, comunidades pequeñas)
pueden mantener la igualdad ya que es una situación matemáticamente
estable
• A altas densidades de población (ciudades…), cualquier fluctuación en una
situación inicial igualitaria se amplificará, generando jerarquía en el estado final
… el comunismo es matemáticamente inestable…
EXAMPLE V: JAMMING TRANSITIONS (TRAFFIC AND DELAY
PROPAGATION)
CRITICAL POINT
UNCERTAINTY
IS MAXIMAL
1/F NOISE AT
CRITICALITY
Lucas Lacasa, Miguel Cea, Massimiliano Zanin Physica A 388 (2009)
EXAMPLE VI: ALGORITHMIC PHASE TRANSITIONS (COMPUTER
SCIENCE AND MATHEMATICS)
Similar to minimize a free energy
THE PRIME NUMBER GENERATOR:
‘artificial chemistry model’
THE PRIME NUMBER GENERATOR:
‘artificial chemistry model’
THE PRIME NUMBER GENERATOR:
‘artificial chemistry model’
500
25
25
20
THE PRIME NUMBER GENERATOR:
‘artificial chemistry model’
500
26
26
500
Positive reactions tend to produce prime numbers
Rate of primes in the steady state ??
M = 10.000
Random presence of
primes 1/log(N)
P = probability that the whole set of numbers become
primes when the algorithm reaches the steady state.
Well defined order
parameter
Characteristics of Criticality
•
•
•
•
Divergence of the correlation length ξ
Certain observables (e.g. distribution of patch sizes) obey power laws
1/f noise (Spectral density)
Universality – extremely different systems display the same behavior
regardless of their dynamical rules.
• System is often sensitive to small perturbations.
• However, criticality is usually obtained by finely tuning a parameter (e.g.
temperature for phase transitions), so they would be unlikely to naturally
arise !!!
However, criticality appears in many places, including:
And hints of criticality appear in many many other places, including:
 Speech statistics
 Music statistics
 Technological evolution (Internet)
 Astrophysics (solar flares)
 Distribution of river basins
 Stock market
 Physiology (EEG, EEC)
… HOW? (very unlikely by a fine tuning process)