Functional differential equations (FDEs) and entropy

C. K. Raju

Honorary Professor
Indian Institute of Education

ckr@ckraju.net

Physics Dept., Mumbai Univ., Kalina Campus, 16 March 2024, 12 noon.

Intro: what are FDEs?

  • Let us start with what you know: ODEs.

Ordinary differential equations (ODEs)

  • Every physics student knows ODEs.
  • A force proportional and opposite to displacement \(y\) leads to
  • the ODE of simple harmonic motion (primes denote derivatives):
\begin{equation} y'' = -k^2y. \label{ODE} \end{equation}

Initial data

  • The 2nd order ODE (1) has a unique solution
  • if we prescribe initial data

  • (initial displacement and initial velocity)

    \begin{align} y(0) &= y_0 , \nonumber\\ y'(0) &= y_1. \label{initial} \end{align}

Time reversibility

  • Above system can be solved forward in time from initial data
  • or backward in time from final data
  • or in any direction from intermediate data.

Determinism

  • Prescription of state ("initial data") at any point of time
  • fully determines the solution (state) for all past and future times.
  • Stateof the world at one instant determines all past and future (on Newtonian physics).

System of \(n\) particles (gas in a box)

  • The above considerations generalize to a system of \(n\) particles in 3D space
  • results in \(3n\) second order ODEs
  • or \(6n\) first-order ODE's
  • state ("initial data") is \(6n\) numbers: \(3n\) positions, and \(3n\) momenta.

Phase space

  • These \(6n\) numbers prescribe a single point in
  • \(6n\) dimensional space called phase space.
  • Uniqueness of solutions means
  • there is a unique trajectory through each point of phase space.
  • All this changes with FDEs.

Functional differential equations (FDEs)

  • Simplest FDE is
\begin{equation} y'(t) = y (t - 1) . \label{RFDE} \end{equation}
  • this is also called a retarded FDE or a delay differential equation
  • the value of \(y'\) is related to PAST values of \(y\).

Insufficiency of initial data

  • Suppose we try to solve the above equation like we solve an ODE.
  • Symbolically
\begin{align} y (1) &= y(0) + \int_0^1 y'(t) dt \\ &= y(0) + \int_0^1 y(t-1) dt. \label{int} \end{align}
  • To actually carry out the integration, we need to know the valus of \(y(t-1)\) between 0 and 1.
  • That is, we need to know \(y(t)\) between -1 and 0, i.e., we need past data
  • a whole function \(\phi\) over the relevant past interval [-1, 0].

Example 1: different pasts, different solns

  • even if "initial data" same.
  • Makes sense intuitively: an FDE relates rate of change NOW to something in past.
  • So, changing past will change solution.

Irreversibility of retarded FDEs

  • Retarded FDEs can only be solved forward in time
  • unlike ODEs they cannot be solved backward in time

Example 2: multiple pasts, same future

  • Consider the FDE \[ y ' (t) = b(t) y(t-1) , \]
  • where
\begin{equation} b(t) = \begin{cases} 0 & t \le 0 , \\ -1 + \cos 2 \pi t & 0 \le t \le 1 , \\ 0 & t \ge 1 . \end{cases} \end{equation}
  • Note that
\begin{equation} \int_{-\infty}^{\infty} b(t)~dt = -1 + \int_0^1 \cos 2 \pi t ~dt = -1 . \label{b} \end{equation}
  • since indefinite integral is \(\frac{\sin 2π t}{2π}\) which vanishes at 0 and 1.
  • Now since \(b(t)=0\) for \(t \le 0\), the FDE is just the ODE \(y'(t)=0\) ,
  • so, for \(t \le 0\), solution is \(y(t) = k\) for some constant \(k\) (= \(y(0)\)).
  • Now, for \(t \in [0,~1]\),
\begin{align} y(t) &= y(0) + \int_0^t y'(s) ds \nonumber \\ \; &= y(0) + \int_0^t b(s) y(s-1) ds \nonumber \\ &= y(0) + y(0) \int_0^t b(s) ds, \end{align}
  • since \(y(s-1) \equiv k = y(0)\) for \(s \in [0, 1]\).

Hence,

  • \(y(1) = 0\), no matter what \(k\) was.
  • But, since \(b(t)=0\) for \(t \ge 1\),
  • the FDE again reduces to the ODE \(y' (t) = 0\), for \(t \ge 1\),
  • so that \(y(1) = 0\) implies \(y(t) = 0\) for all \(t \ge 1\).

The solutions: many pasts same future

No backward solution

  • Hence, the past of a system modeled by a retarded FDE
  • cannot be retrodicted from a knowledge of the entire future.
  • Suppose the future data (i.e., values of the function for \textit{all} future times \(t \ge 1\))
  • are prescribed using a function \(\phi\)
  • if \(\phi\) is different from 0 on \([1,~ \infty ]\),
  • then the retarded FDE admits no backward solutions for \(t \le 1\).
  • If, on the other hand, $φ ≡ 0 $ on \([1,~ \infty ]\),
  • then there are an infinity of distinct backward solutions.
  • In either case, knowledge of the entire future furnishes no information about the past
  • of a system modeled by retarded FDEs.

Information-theoretic entropy

Also known as Shannon entropy

  • Lack of information measured by average number of yes-no question you must ask to determine truth

  • E.g. what is the missing letter?

    p ? t
  • The missing letter must be a vowel,so the word must be one of pat, pet, pit, pot, put
  • Suppose "u" is excluded, and others are equally likely (probability \(pᵢ = \frac{1}{4}\)).

Questioning strategy

  • 1. Is it one of O or I?
  • Then follow the tree.
  • 2 questions are needed
  • \(2 = -\log₂ \frac{1}{4} = - \sum_{i=1}^{4} {}~ pₜ \log pᵢ\)

Relation to Gibbs and Boltzmann entropy

  • Note that information-theoretic or Shannon entropy is
  • similar to Gibbs entropy (up to a multiplicative constant)

\[S_G(ρ) = -k ∫_{R^{6n}}ρ(x) \ln ρ(x) dx\]

  • and Boltzmann entropy \(S_B = k \ln W\) can be seen as a special case of Gibbs entropy.
  • Note that in Gibbs case, \(\rho(x)\) is a probability density
  • which applies to an ensemble (multiple mental copies of the system) in phase space.
  • In Boltzmann case, \(W\) is number of "microstates" consistent with a given "macrostate".
  • Both definitions are a bit vague (which ensemble?, what is a microstate?)

Irreversibility of retarded FDEs and entropy increase

  • Solvability of retarded FDEs from past data
  • + irreversibility (impossibility of solving retarded FDEs backward in time)
  • means on retarded FDE model we have MORE information about past than about future.
  • Means on retarded FDE model entropy increases towards future.

Where do FDEs arise in physics?

  • ODEs part of Newtonian physics: force = mass× acceleration leads to ODEs
  • Continuum version (e.g. fluid dynamics, Navier-Stokes equations) leads to partial differential equations (PDEs).
  • But where are the FDEs?
  • Simplest answer: in the classical electrodynamic many body problem.

This answer has been very hard to explain

  • Discussions in Pune Univ. canteen with colleagues in 1980's
  • Pramod Joag, Naresh Dadhich etc.
  • On issue of time in physics.
  • Understanding time is tricky. Failed to get across my point of view in YEARS.

Early articles

  • Hence, wrote a series of 10 articles in Physics Education
  • UGC journal brought out by Physics Department, Pune U.
  • since the aim was to "carry on the discussion" with people around me
  • and foreign journals were expensive, not easily available.

But people murmured

  • You are talking about big things, changing physics, challenging Einstein
  • but publishing in a local Journal.
  • Valid science decided NOT by experiment/observation
  • but by PUBLICATION by a "prestigious" (=Western) publisher!
  • Science about prestige? Not truth?

Key point of book

  • Physics needs to be reformulated using FDEs (mixed-type)
  • which explains most puzzling qualitative features of QM.
  • Even just retarded FDEs are history-dependent
  • i.e., need past data not just initial data
  • Hence, doing physics using just retarded FDEs involves a paradigm shift in physics.

Groningen debate in 1998

  • Meeting (on "Retrocausality day")was stalled after I claimed paradigm shift due to FDE
  • H. D. Zeh (Heidelberg): Why are FDEs needed in electrodynamics?
    • Heaviside-Lorentz force law (ODE) + Maxwell's equations (PDEs) enough
    • both need only "initial data"
  • Zeh: Hence, no history dependence in physics, no paradigm shift.

BUT my FDEs used the SAME physics

  • with the particle picture.
  • And I said, I SOLVED the FDEs
  • using past data.
  • Challenged Zeh to solve them differently.
  • (He said getting solution was a mathematician's job!)

Different conclusions from same physics?

  • Q1. Are FDEs needed or not?
  • Q2. Does physics need past data?
  • So, let us re-examine Zeh's arguments.

Heaviside-Lorentz force

  • A charge moving in an electromagnetic field moves on Heaviside-Lorentz force
\begin{equation} \vec{F} = q (\vec{E} + \vec{v} \times \vec{B}) . \label{H-L} \end{equation}
  • Motion of charge determined by applying Newton's law of motion (and solving ODEs).

Maxwell's equations

  • The fields \(\vec{E}, \vec{B}\) due to motion of each charge determined using Maxwell's equations:
\begin{align} \nabla \cdot \vec{E} &= \frac{\rho}{\epsilon_0} , \nonumber \\ \nabla \cdot \vec{B} &= 0 , \nonumber \\ \nabla \times \vec{E} &= - \frac{\partial B}{\partial t} , \nonumber \\ \nabla \times \vec{B} &= \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial E}{\partial t} . \label{Max} \end{align}

Solving Maxwell's equations

  • We must solve these PDEs for each charge.
  • Fields \(\vec{E}\) and \(\vec{B}\) are calculated as the derivatives of a scalar potential \(V\) and a vector potential \(\vec{A}\).
\begin{align} \vec{E} &= -\nabla V - \frac{\partial \vec{A}}{\partial t} \nonumber \\ \vec{B} &= \nabla \times \vec{A} \end{align}

Gauge condition

  • Middle two of Maxwell's equations are automatically satisfied.
  • The choice of potential is not unique, and we can add an extra condition (called a gauge condition) to simplify the remaining two equations.
  • In the Lorenz gauge, the first and the last of Maxwell's equations turn into inhomogeneous wave equations
  • for the scalar and vector potential respectively.

Retarded and advanced potentials

  • The solutions are called Lienard-Wiechert potentials
  • The retarded L-W potentials are given by the expressions:
\begin{align} V ( \vec{r} , \: t ) &= \left. \frac{1} {4 \pi \epsilon_0 } \: \frac{qc} {( R c - \vec{R} \cdot \vec{v} )} \: \right |_ {\rm ret} , \nonumber \\ \vec{A} ( \vec{r} , \: t ) &= \left. \frac{\vec{v}} {c^2} \: V ( \vec{r} , \: t) \right |_ {\rm ret}. \label{L-W} \end{align}

Retarded time

  • Here, \(\vec{R} = \vec{r} - \vec{r}_p(t_r)\), where \(\vec{r}_p (t_r)\)
  • denotes the position of the charge \(q\) at the retarded time \(t_r\).
  • The subscript ``ret'' emphasizes that \(\vec{v}\), the velocity of the charge $q$ is also to be evaluated at retarded time \(t_r\),
  • so that \(\vec{v} =\dot{\vec{r}}_p(t_r)\).

Retarded time (contd.)

\begin{equation} c^2(t-t_r)^2 = R^2, \label{retarded} \end{equation}
  • Similarly there are advanced potentials based on advanced (or future) times)

Zeh's foolishness

  • Examining the above formula immediately shows foolish mistake made by Zeh
  • Professor of Physics at Heidelberg and on editorial board of Foundations of Physics
  • "Initial data" for Maxwell's equations means fields \(\vec{E}\), \(\vec{B}\) must be prescribed for one instant of time
  • across all space.

But to determine fields using retarded L-W potentials

  • we need to know (retarded) positions and (retarded) velocities of both charges
  • for ALL past time.
  • So, past data needed.

Groningen debate RESOLVED by my 2004 publication

FDEs = COUPLED ODES + PDES.

  • Coupling essential for 2-body problem of classical electrodynamics
  • because motion of charge 1 depends on fields generated by motion of charge 2
  • and vice versa.
  • Little studied; not found in stock electrodynamics texts, Jackson etc.

My next book on time Eleven Pictures of Time

  • Explained history dependence and tilt (QM) for a layperson.
  • Also explained how Christian church altered beliefs about time to suit its political needs.
  • And how these religio-political belief creep into current science through Stephen Hawking etc.
  • (Won't discuss it today; needs deeper knowledge of formal math and general relativity.)
  • (But if valid science decided by blind belief in social prestige and myth
  • that is a superstition which can be exploited.)

Einstein

  • Also, explained why Einstein did not understand that relativity forces use of FDEs
  • which Poincaré, the true originator of relativity understood.
  • Openly called Einstein a habitual plagiarist
  • This shocked many people caused a major distraction for FDEs.

Truthful science = social disaster

  • Einstein is god, can't be critical of him.

This is an act of heresy bigger than Galileo's

  • (for science is about worshipping scientists about whom there are big myths,
  • not about critically understanding scientific theories).
  • Publicized my viewpoint, e.g. 2004 news report.

2005 was centenary of Einstein's 1905 special relativity paper

  • titled "On the electrodynamics of moving bodies".
  • Michael Atiyah, former President, Royal Society, Abel Laureate,Fields Medalist gave his "Einstein Centenary Lecture"
  • in which he repeated my point: FDEs can perhaps explain QM

  • but added, "Don't forget I suggested this"
  • To drive point home, in a socially savvy way
  • he got it named "Atiyah's hypothesis", not "Einstein's mistake"

Later forced to admit my priority

Some other interim publications

Then a series of articles on FDEs

  • again in Physics Education
  • Retarded Gravitation Theory 2.0 was developed during Covid but not published.
  • Why not? Because I am busy changing math education (impacts more people)
  • and physicists may take very long to correct their time beliefs.

Summary of background

  • Church politics of time (and related dogmas) have crept deep into Western thought AND science
  • because of (1) long-term church hegemony over West (till 19th c.)
  • and(2) Western hegemony over us and science after colonialism.
  • Hence, conflicts about time in physics are hard to resolve.

Thermodynamics

Boltzmann

  • Conflicts about time dogged 19th c. physics
  • especially thermodynamics
  • resulting in Boltzmann's suicide in 1906.

Second law of thermodynamics

  • Entropy is non-decreasing.
  • If \(S\) is entropy, \(\Delta S ≥ 0\)
  • Boltzmann tried to prove H-theorem
  • Entropy INCREASES until it reaches a maximum in state of thermodynamic equilibrium.

Problem: SLT incompatible with Newtonian physics

  • Gas in a box is a collection of \(n\) molecules
  • colliding elastically with each other and with walls of container.
  • Elastic collisions means energy is conserved.
  • Molecules motions obey Newtonian physics.

Reversibility paradox (Loschmidt)

  • Solution of ODEs is reversible
  • so if entropy increase towards future
  • it must also increase towards past
  • Hence, entropy must stay constant.

Recurrence paradox

  • For formal math treatment see Time book, chp. 4, and appendix, and this explanation and this one.
  • Here we give only the usual intuitive account in physics texts.
  • Through every point of phase space there is a unique trajectory
  • so trajectories in phase space never intersect.

Since exact position in phase space is not known

  • imagine a swarm of points flowing in phase space.
  • Since phase space trajectories never intersect this swarm behaves like the flow of an incompressible fluid.
  • That is, flow in phase space preserves volume (Liouville's theorem).

Recurrence paradox

  • For a gas in a box with finite energy phase space volume is finite
  • therefore, the flow must return arbitrarily close to its initial state
  • hence infinitely often.(Poincaré recurrence theorem.)
  • Therefore, entropy cannot increase.

Recurrence paradox also applies to macro states

  • E.g., if the gas is initially in one half of the box
  • this state will recur repeatedly.

Use of FDEs instantly resolves both paradoxes

  • Retarded FDEs are irreversible, hence reversibility paradox is resolved
  • Retarded FDEs exhibit phase collapse, hence volume NOT preserved in phase space
  • Hence, recurrence theorem does NOT apply.

Interim summary

  • With retarded FDEs entropy increase perfectly possible
  • Boltzmann need not have committed suicide.

So what have be achieved?

  • Explained entropy increase by assuming retarded potentials
  • Derived thermodynamic arrow of time from electrodynamic arrow of time.

No assumption here

  • We can restore time asymmetry by considering advanced solutions of wave equation.
  • These result in advanced FDEs which will reduce entropy.
  • Are these solutions physical?
  • Yes, see paper on time travel. Will result in spontaneity (shown by living organisms)

Tilt

  • Most general case is that of mixed-type FDEs (with a "tilt")
  • a convex combination of retarded and advanced propagator, where retarded interactions dominate.
  • Domination of retardediteractions is an observation, not an assumption.

Where are the FDEs in thermodynamics?

  • FDEs arise naturally in electrodynamics and relativity
  • (Atiyah's claim that this is a "hypothesis" is foolish)
  • But how is that relevant to thermodynamics or to a (non-relativistic) gas in a box?

Molecular interactions

  • A gas consists of molecules which consist of atoms
  • which have a positively charged nucleus
  • surrounded a by a negatively charged electron cloud
  • An atom/molecule is electrically neutral only as an approximation at large-distance from its nucleus.
  • At small distances, it may involve induced or permanent dipole, quadrupole, multipole interactions.

This results in short-distance van der Waals forces

  • which are both attractive and repulsive (because electron clouds repel each other).
  • Keesom force (perm dipole- perm dipole)
  • Debye force (perm dipole-induced dipole)
  • London force (fluctuating dipole-induced dipole)

Lennard-Jones inter-molecular 6-12 potential

\[ V = \frac{A}{r^{12}} - \frac {B}{r^6} \]

  • \(r\) is distance between two particles
  • (\(A = 4ε \sigma^{1/12}\), \(B = 4ε \sigma^{1/6}\)

Use of "force" (instantaneous action at a distance) means use of ODEs,

  • since 19th c. physics was about Newtonian physics
  • But the molecules in question are moving
  • so molecular interactions should involve retarded action at a distance between dipoles etc.
  • Modelling via retarded FDEs will add a small velocity-dependent component but explain entropy increase.

Concluding remarks

  • Modelling molecular motion via retarded FDEs
  • cleanly resolves the reversibility and recurrence paradoxes of thermodynamics.

Future research

  • Modelling via mixed-type FDEs (with a "tilt in the arrow of time")
  • allows spontaneity at the level of biological macromolecules
  • hence to biological organisms.
  • That is the next big frontier.

Additional reading: relation of science to religion

Note