Introduction to Statistical Thermodynamics of Soft and Biological Matter Dima Lukatsky FOM Institute for Atomic and Molecular Physics [AMOLF], Amsterdam Acknowledgements: Samuel Safran, Weizmann Institute of Science Daan Frenkel, FOM-AMOLF

References Statistical Thermodynamics of Surfaces, Interfaces and Membranes S. A. Safran, (Addison Wesley, 1994). Biological Physics. Energy, Information, Life Philip Nelson, (Freeman and Company, New York, 2004). Understanding Molecular Simulations D. Frenkel and B. Smit (Academic, 2002). Intermolecular and Surface Forces J. Israelachvili, (Academic, 1992). The Colloidal Domain: Where Physics, Chemistry, Biology and Technology Meet F. Evans and H. Wennerstrom, (Wiley, 1994).

Main Questions and Ideas How can living organisms be so highly ordered ? Equilibrium versus non-equilibrium systems. Living systems are not at equilibrium, and they are open. Quasi equilibrium. Interactions can lead to a spontaneous ordering even at equilibrium. Entropy can lead to a spontaneous ordering at equilibrium ! Flow of information characterizes living organisms. Evolution is the biological “pressure”. Living organisms are robust.

Biology is living soft matter Self-assembly High specificity Multi-component Information

Thermal energy and molecular length-scale - characteristic energy scale - Boltzmann constant DNA 2 nm 25 nm microtubule E. coli 1  m = 1000 nm Bacteriophage virus bp DNA 100 nm

Statistical description of random World The collective activity of many randomly moving objects can be effectively predictable, even if the individual motions are not. If everything is so random in the nano-world of cells, how can we say anything predictive about what’s going there ? M.W. Davidson, FSU

Entropy. The 2 nd law of thermodynamics Isolated system always evolve to thermodynamic equilibrium. In equilibrium isolated system has the greatest possible ENTROPY (disorder*) allowed by the physical constraints on the system. * Sometimes, high entropy means more order!

Entropy as measure of disorder Number of allowed states in A: Number of allowed states in B: Number of allowed states in joint system A+B: Entropy: Entropy is additive:

How to count states Total number of states: Using Stirling formula: Probability of each state: Molecules A Molecules B Entropy of system:

Entropy… Probability of each state: Molecules A Molecules B

Sequence Analysis Course… Lecture 9 Shannon definition INFORMATION ENTROPY

Entropy of ideal gas Indistinguishablility For N molecules: For one molecule: - “cell” volume (quantum uncertainty ) V – total volume Free energy of ideal gas: density:

So what IS entropy?

Ludwig Boltzmann The Entropy is equal to ….

Ludwig Boltzmann The logarithm of the number of states …

Ludwig Boltzmann Times My constant!

During a spontaneous change in a closed system, the DISORDER increases …. WATCH OUT! FAKE Second Law

Ordered Solid Disordered Liquid

Hard-sphere crystal Hard-sphere liquid Hard-sphere freezing is driven by entropy ! Lower Entropy… Higher Entropy…

Colloidal “Entropic” crystal OPAL

Entropy and Temperature System A System B Total energy: Isolated (closed) system: Number of allowed states in A Total number of allowed states Total entropy

Entropy Maximization System A System B A and B in thermal contact. Total system A+B is isolated. Total energy: Total entropy: Define Temperature:

Ordering and 2 nd law of thermodynamics - Condensation into liquid (more ordered). - Entropy of subsystem decreased… - Total entropy increased! Gives off heat to room. System in thermal contact with environment Equilibration Initially high Cools to room

Free energy Reservoir, T Small system For closed system: a For open (small) system Free energy is minimized: - Helmholtz free energy - Gibbs free energy