Friday, July 15, 2011

The leibniz

In order to motivate the study of thermal physics, Chapter 3 of the 4th edition of Intermediate Physics for Medicine and Biology begins with an examination of how many equations are required to simulate the motion of all the molecules in one cubic millimeter of blood. Russ Hobbie and I write
“It is possible to identify all the external forces acting on a simple system and use Newton’s second law (F = ma) to calculate how the system moves … In systems of many particles, such calculations become impossible. Consider, for example, how many particles there are in a cubic millimeter of blood. Table 3.1 shows some of the constituents of such a sample [including 3.3 × 1019 water molecules]. To calculate the translational motion in three dimensions, it would be necessary to write three equations for each particle using Newton’s second law. Suppose that at time t the force on a molecule is F. Between t and t + Δt, the velocity of the particle changes according to the three equations

vi(t+Δt) = vi(t) + FiΔt/m, (i = x, y, z).

The three equations for the change of position of the particle are of the form x(t + Δt) = x(t) + vx(t)Δt … Solving these equations requires at least six multiplications and six additions for each particle. For 1019 particles, this means about 1020 arithmetic operations per time interval … It is impossible to trace the behavior of this many molecules on an individual basis.

Nor is it necessary. We do not care which water molecule is where. The properties of a system that are of interest are averages over many molecules: pressure, concentration, average speed, and so forth. These average macroscopic properties are studied in statistical or thermal physics or statistical mechanics.”
It is difficult to gain an intuitive feel for just how many differential equations are needed in such a calculation, just as it is difficult to imagine just how many molecules make up a macroscopic bit of matter. Chemists have solved the problem of dealing with large numbers of molecules by introducing the unit of a mole, corresponding to Avogadro’s number (6 × 1023) of molecules. Other quantities involving Avogadro’s number are similarly defined. For instance, the Faraday corresponds to the magnitude of the charge of one mole of electrons (I admit, the Faraday is more of a constant than a unit); see page 60 and Eq. 3.32 of Intermediate Physics for Medicine and Biology. In Problem 2 of Chapter 14, Russ and I discuss the einstein, a unit corresponding to a mole of photons. When doing large-scale numerical simulations on a computer, it would be useful to have a similar unit to handle very large numbers of differential equations, such as are required to model a drop of blood.

Fortunately, such a unit exists, called the leibniz. Sui Huang and John Wikswo coined the term in their paper Dimensions of Systems Biology, published in the Reviews of Physiology, Biochemistry & Pharmacology (Volume 157, Pages 81-104, 2006). They write
“The electrical activity of the heart during ten seconds of fibrillation could easily require solving 1018 coupled differential equations (Cherry et al. 2000). (N.B., Avogadro’s number of differential equations may be defined as one Leibnitz, so 10 s of fibrillation corresponds to a micro-Leibnitz problem.) Multiprocessor supercomputers running for a month can execute a micromole of floating point operations, but in the cardiac case such computers may run several orders of magnitude slower than real time, such that modeling 10 s of fibrillation might require 1 exaFLOP/s×year.”
The leibniz appeared again in Wikswo et al.’s paper Engineering Challenges of BioNEMS: The Integration of Microfluidics, Micro- and Nanodevices, Models and External Control for Systems Biology in the IEE Proceedings Nanobiotechnology (Volume 153, Pages 81-101, 2006).
“What distinguishes the models of systems biology from those of many other disciplines is their multiscale richness in both space and time: these models may eventually have millions of dynamic variables with complex non-linear interactions. It is conceivable that the ultimate models for systems biology might require a mole of differential equations (called a Leibnitz) and computations that require a yottaFLOPs (floating point operations per second) computer.”
If we take the leibniz (Lz) as our unit of simulation complexity, the calculation Russ and I consider at the start of Chapter 3 requires solving approximately 6 × 1019 differential equations, or about 0.1 mLz. Note that we describe two first order differential equations for each molecule, but others might prefer to speak of a single second-order differential equation. This would make a difference of a factor of two in the number of equations. I propose that when using the leibniz we consider only first order ODEs. Moreover, when using a differential equation governing a vector, we count one equation per component.

For those not familiar with Gottfried Leibniz (1646 – 1716), he is a German mathematician and a co-inventor of the calculus, along with Isaac Newton. In fact, Leibniz and Newton got into one of the biggest priority disputes in the history of science about this landmark development. Newton has his unit, so its only fair that Leibniz has one too. Leibniz also made contributions to information theory and computational science, so the liebniz is a particularly appropriate way to honor this great mathematician.

John Wikswo, my PhD advisor when I was in graduate school at Vanderbilt University, notes that there are two alternative spellings of Leibniz's name: Leibnitz and Leibniz. I favor “Leibniz”, the spelling on Wikipedia, and so does Wikswo now, but he points out that there’s plenty of support for “Leibnitz” used in his earlier publications. I had high hopes of enjoying a bit of fun at my friend’s expense by adding an annoying “[sic]” after each appearance of “Leibnitz” in the above quotes, but then Wikswo pointed out that Richard Feynman used “Leibnitz” in the Feynman Lectures on Physics. What can I say; you can’t argue with Feynman.

1 comment:

  1. No, you can not argue with Feynman, but wouldn't it be nice to have some time at the board with him. Toe to toe, with the Master!