Statistical physics. Emmanuel Trizac LPTMS / University Paris-Sud
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1 Statistical physics Emmanuel Trizac LPTMS / University Paris-Sud
2 Outline I Introduction to phase transitions and critical phenomena 1- The problems raised by phase transitions, from a statistical mechanics perspective 2- Classification of phase transitions 3- The drosophila of phase transitions 4- Order parameter and symmetry breakdown 5- Local order and correlation functions : from magnets to liquids II First order phase transitions 1- Unstable isotherms, double-tangent and Maxwell construction 2- Spinodal and binodal 3- Changing ensembles 4- van der Waals equation 5- The case of mixtures III Critical phenomena : qualitative approaches 1- Weis molecular field 2- Variational mean-field 3- Critical exponents 4- Landau theory 5- Correlation functions and Ginsburg-Landau functional 6- Validity of mean-field
3 Statistical mechanics : considered by some as one of the most intellectually challenging subject. Noted for a high suicide rate among founders
4 Prerequisites Elementary statistical mechanics Honest person toolbox in mathematics Linear algebra Probability theory Complex analysis 2 2 (x 1)( x 2 i) f (x)= 2 x +2+2 i
5 Introduction to phase transitions and critical phenomena 1- Problems raised by phase transitions, from a stat mech perspective 2- Classification of phase transitions 3- The drosophila of phase transitions 4- Order parameter and symmetry breakdown 5- Local order and correlation functions : from magnets to liquids
6 Surfactant molecules and micelle formation Basic mechanism Opens for considerable richness
7 Various surfactant phases (a) An amphiphilic molecule (b) Spherical micelle (c) Cylindrical micelle (d) Cubic phase (e) Inverse micelle (f) Hexagonal phase (g) Bicontinuous cubic structure (h) Vesicle (i) Lamellar phase (j) Sponge phase
8 Identify an ORDER PARAMETER pressure density
9 Complexity / variety of phase changes
10 Non spherical colloids...
11 Phase behaviour of hard sphero-cylinders P. Bolhuis, 1996
12 Introduction to phase transitions and critical phenomena 1- Problems raised by phase transitions, from a stat mech perspective 2- Classification of phase transitions 3- The drosophila of phase transitions 4- Order parameter and symmetry breakdown 5- Local order and correlation functions : from magnets to liquids
13 Examples and exceptions Water Generic case dp 1 Lliq gas ( coex)= dt T v gas v liq
14 What about magnets? magnetization magnetic field
15 MAGNETS
16 Analogy between magnets and liquids? density magnetization pressure magnetic field
17 Definition of critical exponents fluids magnets
18 Why are critical exponents interesting? several layers of universality E. A. Guggenheim, J. Chem. Phys. 13, 253 (1945) Corresponding states for liquid-gas transition Tc (K) Pc (atm) Ne Ar Kr Xe N O CO CH
19 There is more to universality Liquid-gas transition: Magnets with uniaxial anisotropy (MnF2): Phase separation in binary mixture (CCl4+C7F16): 3d Ising model on cubic lattice, fcc etc : all belong to the same universality class What matters is space dimension + symmetry of order parameter e.g. for Ising in 2D :
20 Introduction to phase transitions and critical phenomena 1- Problems raised by phase transitions, from a stat mech perspective 2- Classification of phase transitions 3- Ising model : the drosophila of phase transitions 4- Order parameter and symmetry breakdown 5- Local order and correlation functions : from magnets to liquids
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