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# SISTEMAS COMPLEJOS TEMA II – LA FÍSICA DE LOS SISTEMAS COMPLEJOS.

## Presentación del tema: "SISTEMAS COMPLEJOS TEMA II – LA FÍSICA DE LOS SISTEMAS COMPLEJOS."— Transcripción de la presentación:

SISTEMAS COMPLEJOS TEMA II – LA FÍSICA DE LOS SISTEMAS COMPLEJOS

STATISTICAL MECHANICS in a nutshell Why stat.mech.? Description of gases with Newtons 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

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 FUNDAMENTAL CONCEPT IN STAT. MECH.: ensemble!

EXAMPLE 1 System: N 2 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 …

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 EXAMPLE 2

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 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

FROM THE PARTITION FUNCTION WE CAN CALCULATE ALL THERMODYNAMICAL QUANTITIES Partition function - Total energy - Free energy - Entropy Microscopic Macroscopic Hamiltonian H Partition function Z Free energy F Entropy S Pressure P Magnetization M … Quantifies interaction strength between elements F F

FROM THE PARTITION FUNCTION WE CAN CALCULATE ALL THERMODYNAMICAL QUANTITIES Partition function - Total energy - Free energy - Entropy - Temperature Microscopic Macroscopic Hamiltonian H Partition function Z Free energy F Entropy S Pressure P Magnetization M … Quantifies interaction strength between elements F F Entropy, disorder, information

FROM THE PARTITION FUNCTION WE CAN CALCULATE ALL THERMODYNAMICAL QUANTITIES Partition function - Total energy - Free energy - Entropy - Temperature Microscopic Macroscopic Hamiltonian H Partition function Z Free energy F Entropy S Pressure P Magnetization M … Quantifies interaction strength between elements F F The inverse of the temperature is the cost of buying energy from the rest of the world. Entropy is the currency being paid. Inverse temperature is the cost in entropy to buy a unit of energy.

FROM THE PARTITION FUNCTION WE CAN CALCULATE ALL THERMODYNAMICAL QUANTITIES Partition function - Total energy - Free energy - Entropy - Temperature Microscopic Macroscopic Hamiltonian H Partition function Z Free energy F Entropy S Pressure P Magnetization M … Quantifies interaction strength between elements F F

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. Lets consider to apply this method to the statistics of living beings, society, sociology and so forth. 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 … Ludwig Edward Boltzmann 1844 - 1906)18441906

WHAT IS A PHASE TRANSITION? OPTIMIZATION PROCESS: 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

Order parameter vs Control parameter CLASSIFICATION OF PHASE TRANSITIONS 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

-No latent heat involved -Order parameter changes continuously at the transition -Other magnitudes diverge EXAMPLE (second order transition) Ferromagnetic-Paramagnetic transition

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 second order phase transition- pitchfork bifurcation. dy/dt = y – by 3, dy/dt = a + by 2

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) Local rules (similar to Newton laws of motion) Develop a statistical mechanics approach Develop a statistical mechanics approach Find different phases and in some cases Find different phases and in some cases Phase transitions 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

1. T=0 well ordered 2. 0Tc disordered Phase transition. Global magnetization PLDs

ISING MODEL (numerical simulations – Monte Carlo) TTc

EXAMPLE II: VOTER MODEL (SOCIOLOGY)

Voter model Voter model spins are people opinion(red, blue)

EXAMPLE III: FLOCKING MODELS (COLLECTIVE PHENOMENA)

Noise-induced transitions Noise-induced transitions: animal flocking Flock1 Flock2

A simple model: Follow your neighbors ! absolute value of the velocity is equal to v 0 new direction is an average of the directions of neighbors plus some perturbation η j (t)

EXAMPLE IV: BONABEAU MODEL (SOCIOLOGY)

(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. Social hierarchy generation: Bonabeau model with 2 opposite mechanisms

Si i gana, intercambia la casilla con j. Si pierde, se mantienen las posiciones. Después de cada enfrentamiento los status h i (t) y h j (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

Lucas Lacasa, Miguel Cea, Massimiliano Zanin Physica A 388 (2009) UNCERTAINTY IS MAXIMAL 1/F NOISE AT CRITICALITY

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 20

THE PRIME NUMBER GENERATOR: artificial chemistry model Positive reactions tend to produce prime numbers 500 26 500

Rate of primes in the steady state ?? Random presence of primes 1/log(N) M = 10.000

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)

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