Time Reversibility

Notes from .

Introduction

• “Microscopic” Physical theoreies describe motion, predicting how events will unfold with the passage of time.
• Physical theoreis replace physical description of nature by set of coupled differential equations
  • these equations give explicit time derivatives for interesting variables in terms of present value
  • As time is continous, numerical solution of such equations need very small time step.
• Newton, “I do not define time, space, place, and motion, since they are well known to all.”
• All classical phyiscal theories like Newton’s, Schrodinger’s and Maxwell’s do not display any explicit distintion between the future and the past.
  • This means they are time reversible.
  • Solving the equation forward in time will give the same solution as solving it backward in time.
• However, what about the one-way arrow of time?
  • Our macroscopic experiences are filled with irreversible processes.
  • These “dissipative” processes can be summarized in “irreversible thermodynamics”.
  • If any “information” is degraded into heat or lost, the ability to recapture the past is also lost.
  • Order of the past has been replaced with relative disorder
• Irreversibility: loss of information required to describe or recover the past.
  • Lost information makes it impossible to reverse the flow.
• Mechanics coupled with boundary conditions, constraints and driving forces can describe irreversible processes
  • Call this “thermomechanics”
• Lyapunov instability -> chaos
  • sensitivity to initial conditions
• Engineering problems include irreversible phenomena like viscosity, heat conductivity, plasticity, etc.
  • Consider continuum theories (FEM) and macroscopic variables which are averaged quantities like temperature, strain rate, and stress.
  • Also, systems considered are not isolated, external sources and sinks for heat and work are included.
  • Statistical averaging is important to get the irreversibility from reversibly dynamics
• Macro Thermodynamics:
  • Differs from mechanics in 3 ways:
    1. System states include thermal variables, but neglect fluctuations in mechanical or thermal vairbales.
    2. Mechanisms for changes are omitted too, thermodynamics describes which states are possible, but does not predict transformation rates or mechanisms.
    3. System description is macroscopic and less detailed.
  • Main use of the thermodynamic view is what is possible.
    • Laws of thermodynamics show it is illegal to have temp drop with added head
  • Sometimes reffered to as “thermostatics” as the dynamic mechanisms are omitted.
    • The states are temperature and entropy.
    • Focus on entropy!!!
      • Gibbs entropy
      • Shannon’s information
      • Maximizing entropy corresponds to minimal information.
      • Lyapunov instability
        • Maps, attractors, Lyapunov Spectra
      • Bit-reversibility
      • Fractals are universal feature of irreversibility.
  • Dissipative systems: open systems which lose energy to the enviornment.
    • Surroundings acts as sources and sinks for momentum and energy, and the “information” generated by chaos.
    • Rayleigh-Benard problem to demonstrate this problem.
    • Forward evolution seeks out relatively stable attractors
    • Reverse record of this evolution corresponds to an unstable unobservable repeller.
  • Irreversability is a consequnces of nonlinearity, chaos, mixing and bifurcations.
    • Without chaos and mixing, no possibility of memory loss
    • Attractors destroy information which is required to reverse them.