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Biomedical Physics with Applications to Disease (BPAD)
Chapter 2: Foundations of Physics and Mechanics
BPAD Team + SOCR
March 2024
Earlier, in the mathematical foundations chapter we discussed the concepts of scalars, vectors, matrices, and tensors, as well as some of the corresponding (tensor) operations, e.g., scalar multiplication, vector product, and matrix multiplication.
1 Distances and Sizes
In its most general form, the term distance is defined as the amount of “space” separating two objects, states, or even abstract concepts, where space is the union of all possible objects or states. In geometric settings, distance between two points represents a numerical measurement reflecting the size of space separating the points. In physics, a distance measure characterizes the physical length between particles or objects. For a pair of objects, think about two points \(A\) and \(B\), the distance between them may be denoted by \(dist(A,B)= |AB|\equiv |BA|=dist(B,A)\). The mathematical description of a distance function (metric) describes proximity of elements of some given space - nearby and far-apart objects correspond to small and large distances, respectively.
Examples of distance metrics, i.e., space-metric pairs, include:
- The Euclidean distance between two points in the 2D plane. Pick any pair of points \(P_1=(x_1, y_1),\ P_2=(x_2, y_2)\in \mathbb{R}^2\equiv \mathbb{C}\), then, the distance between them is
\[d(P_1,P_2)=\sqrt {(\Delta x)^2+(\Delta y)^2} =\sqrt {(x_2-x_1)^2+(y_2-y_1)^2}.\] This formulation of a distance in \(\mathbb{R}^2\) is derived by applying the Pythagorean theorem. The result is a 2D flat (Euclidean) metric-space \(\left(\mathbb{R}^2, d\right)\). Note that the distance \(d\) leads to the arc-length distance quantifying the distance between pairs of points along a curve embedded in \(\mathbb{R}^2\).
There are many other distance metrics that can be defined to construct \(n\)-dimensional Euclidean and non-Euclidean metric-spaces. For instance the Euclidean metric-space \(\mathbb{R}^n\) can be equipped with some of the following distance measures \(d_p(X,Y):\mathbb{R}^n\times \mathbb{R}^n \to \mathbb{R},\ \forall p\in \mathbb{R}_{\ge 1}\) (Minkowski distance of order \(p\), or simply \(p\)-norm):
\[d_p(X,Y) =\left(\sum_{i=1}^{n}\left|x_i-y_i\right|^{p}\right)^{\frac{1}{p}}\equiv ||X-Y||_p.\] However, the infinity norm distance, or Chebyshev distance, provides yet another way to track object separation in \(\mathbb{R}^n\): \[d_{\infty}(X,Y)=\lim _{p\to \infty}\left(\sum_{i=1}^{n}\left|x_{i}-y_{i}\right|^{p}\right)^{\frac{1}{p}} \equiv \max \left(|x_1-y_1|,|x_2-y_2|,\cdots ,|x_n-y_n|\right)\equiv ||X-Y||_{\infty}.\]
In biophysical problems, the experimental space, e.g., \(\mathbb{R}^n\), is often equipped with a Euclidean distance of the corresponding dimension, e.g., \(d_{2,\mathbb{R}^n}(\cdot,\cdot)\). This choice provides a natural metric-space to study the phenomenon where length/distances are invariant with respect to rigid-body transformations, e.g., rotation and offset/shift.
All distance measures must satisfy three metric conditions. On the space \(M\), a distance function \(d\) is a map \(d(\cdot,\cdot):M\times M \to \mathbb{R}^+\) must obey the following conditions:
- \(d(x,y) \geq 0,\ \forall x,y\in M\), with \(d(x,y) = 0 \iff x = y\)
- \(d(x,y) = d(y,x),\ \forall x,y\in M\)
- \(d(x,z) \leq d(x,y) + d(y,z),\ \forall x,y,z\in M\).
In the simple situation where \(M\equiv \mathbb{R}\) is the space of the real scalars, the natural definition of the distance between a pair real numbers \(x, y\in\mathbb{R}\) is \(d(x,y) = |x − y|\), the absolute-value of the numerical difference between the numbers. Check that this distance satisfies the three metric conditions. An alternative distance formulation on the same state-space is \(\tilde{d}(x,y) = \begin{cases} 0 & x=y \\ 1 & x\not= y \end{cases}\), which defines a different metric corresponding to another topological configuration of the real line called discrete topology where all reals are dispersed and can never be arbitrarily close to each other (say within \(0.1\) units from one another).
The size of a space, e.g., state space of a biophysics experiment, is measured using a properly defined metric on the space. For instance, length (1D space), area (2D space), volume (3D space), and and hyper-volume (\(n\)D space) represents commonly used quantitative measures to capture the size of a neighborhood, or a subspace/subset, in the original space. Length naturally corresponds to sizes of 1D linear segments, whereas area and volume capture the size of a closed region in a 2D region in the plane and a 3D solid, respectively. For instance, using the classical Euclidean distance, the area of a square in 2D, with side of \(1\ \text{unit}\), is equal to \(1\times 1=1\ \text{unit}^2\) and the volume of a ball of radius \(r\) anywhere in 3D is \(\frac{4}{3}\ \pi\ r^3\).
The FSU Galactic-to-Microscopic animation shows the Universal scale from a distant cosmic view of the Milky Way Galaxy, from a distant viewpoint 10 million light years away from Earth (\(10^{23}\) meters), down to the peculiar world of the quark particles (\(10^{-16}\) meters) by orders of magnitude magnification (\(\times10\)). This universal view starts with relativistic distances and goes through the Newtonian and human scales into the enigmatic microscopic world of biological cells, DNA, electrons, and protons.
Biophysical systems study objects spanning a wide range of sizes, scales, shapes, and forms. In the metric system, the meter is the universal unit of length. Life diversity increases with lowering length scale, i.e., the variety and dynamics of microorganisms at lower scales are much more pronounced than their complex-life-form counterparts at larger scales.
Human vision allows us to observe experiments with naked eyes for scales from around \(1\ km\equiv 10^3\ m\) to about \(1\ mm \equiv 10^{−3}\ m\). Of course, we use instruments to enhance our senses, peer outside of this limited range. Advanced hardware, software, and artificial intelligence allow us to study detailed structure of distant galaxies light-years away (\(\ge 10^{16}\ m\)) or tiny objects \(1\ \mu m\equiv 10^{−6}\ m\) (micrometer, micron), \(10^{−10}\ m\) (nano=-meter), etc.
