Calculating Trends in Natural Sciences
The laws of Classical Physics often apply to smooth changes in analytic processes. Differential Calculus describes them adequately more often than not, since the functions associated with them are continuous and sometimes even twice differentiable. However, the functions describing some of these laws ocassionally incur non-differentiabilities. Moreover, significant modern scientific discoveries increasingly unveil non-smooth physical processes, whom the derivative often describes suboptimally. We will distinguish between two types of functions’ singularities: continuous non-differentiable functions and discontinuous ones. Instead of calculating the rate of change, we can often still describe these functions’ monotony behavior at their singularity points by settling with their “local trend” there. We may obtain it with the new “Detachment” operator. On top of its applicability at singularities, we will explain how the detachment can serve as an auxiliary thought tool in stationary points, where the derivative vanishes, but the process isn’t constant.
1 Introduction
The combination of discrete and continuous mathematics introduces philosophical and pure mathematical merits. For example, Lovasz advocates building bridges between them in [1], and Tao builds such bridges in [4]. However, the merge between these philosophical conceptions is not purely bound to artificial computer applications or Mathematics. Instead, it seems like nature itself unifies the discrete and continuous. For example, Strauss discusses in [3] some examples of scientific phenomena that we view today as discrete and continuous processes simultaneously.
A famous example is the duality of waves and particles. Classical physicists, such as Leibniz, thought that nature “makes no leaps” in the sense that it is continuous and infinitely further divisible. However, the $20^{th}$ century physicists discovered that several properties of nature, such as energy, momentum and position, are discontinuous in small enough scales. Strauss further outlines examples in the Biological realm.
In this article, we claim that the combination of the discrete and continuous is not merely an inclination of nature as Strauss mentioned. We further augment the argument and suggest that such a combination, when applied with the new Detachment operator, often outperforms the derivative in describing some natural phenomena. We bring theoretical physical examples where the derivative, applied at a point, does not fully capture the function’s monotonicity near it, while the detachment does. In part 2 we recall the definition of the detachment operator. In part 3 we bring examples of analyzing functions at stationary points in classical physics as well as in phase diagrams. In part 4 we illustrate non-differentiable functions. We distinguish between isolated singularities, as is the case in many power rules such as Kepler’s law, and nowhere differentiable functions, as in Brownian motion. In part 5 we mention some of the discontinuities that characterize phase transition. In part 6 we conclude with a philosophical discussion about the scientific trends that led to the invention of the detachment operator.
2 Reminder: the definition of the Detachment operator
Recall that the detachment operator is defined as follows.
Definition 1. Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ be a real function. Then its one-sided detachments are defined as follows:
$$\begin{array}{ccc}
& f_{\pm}^{;}:\;\mathbb{R}\rightarrow\left\{ -1,0,+1\right\} \\
& f_{\pm}^{;}\left(x\right)\equiv\pm\underset{{\scriptscriptstyle h\rightarrow0^{\pm}}}{\lim}sgn\left[f\left(x+h\right)-f\left(x\right)\right].
\end{array}$$
For more details, see the second post in this posts series.
3 Functions with a vanishing derivative
To analyze functions’ monotonic behavior around stationary points where their derivative vanishes, we often resort to one of two workarounds: performing a higher-order derivative test, or calculating the first derivative in one-sided neighborhoods of the stationary point. In this section we suggest the detachment as a simple auxiliary thought tool that serves as another workaround.
3.1 Stationary points of moving objects
Consider an object whose position at time $t$ is $x\left(t\right).$ We are often interested in the object’s “instantaneous” trend.
Definition 2. We say that the function trends forward locally from one of the sides at $t_{0}$ if there exists a corresponding one-sided $\delta-$neighborhood of $t_{0}$ such that for any $t^{*}$ there, it holds that $x\left(t^{*}\right)>x\left(t_{0}\right).$ Similarly, it trends backwards locally or freezes if $x\left(t^{*}\right)<x\left(t_{0}\right)$ or $x\left(t^{*}\right)=x\left(t_{0}\right),$ respectively.
If the object’s one-sided velocity, defined by $v\left(t\right)=x’_{+}\left(t\right)$ is well-defined and doesn’t vanish, then the trend is given by $sgn\left[v_{+}\left(t\right)\right].$ Thus, for example, if the object has been moving forward continuously around the point, then the position’s derivative sign will be an adequate tool for properly describing its trend.
However, if the object’s (one-sided) velocity vanishes, its sign may not capture the trend.
Consider, for example, the case where the position and velocities are described by the following equations:
$$\begin{cases}
x\left(t\right)=t^{k},\\
v_{+}\left(t\right)=kt^{k-1}.
\end{cases}$$
Let us restrict ourselves to the natural exponents in this subsection, and treat irrational ones in section 4.1. Thus, assume that $k\in\mathbb{N}^{+}$ is a given constant, and $t\geq0.$ Then the object’s one-sided velocity at $t=0$ is:
$$v_{+}\left(0\right)=\begin{cases}
0, & k\neq1\\
1, & k=1.
\end{cases}$$
The object’s speed depends on the slope of the position’s graph because it measures its rate. However, what about the trend? Isn’t the object trending forward at $t=0,$ for any $k$? Note, however, that the derivative sign vanishes for any $k\neq1.$ Therefore, it doesn’t capture the trend coherently. Sanderson introduces this issue without specifically mentioning trends calculations, in [5].
We may apply the higher-order derivative test and continue differentiating the function $x$ until we arrive at a derivative that doesn’t vanish. Its sign, as well as the order’s parity, are indicative of the local trend with a set of conditions.
However, wouldn’t it be convenient to think of trends more concisely?
Let us calculate the right-detachment of $x$ at $t=0$:
$$\begin{align*}
x_{+}^{;}\left(0\right)&\equiv\underset{t\to0^{+}}{\lim}sgn\left[x\left(t\right)-x\left(0\right)\right]=\underset{t\to0^{+}}{\lim}sgn\left(t^{k}\right)\\&=\underset{t\to0^{+}}{\lim}\left[sgn\left(t\right)\right]^{k}=\left[\underset{t\to0^{+}}{\lim}sgn\left(t\right)\right]^{k}=+1,
\end{align*}$$
where the third transition is due to the multiplicativity of the sign function. Thus, we are able to think of the trend directly and without having to go through higher-order rates. It is easy to show, as we do in Semi-discrete Calculus, that definitions 1 and 2 are identical. For more details, see the fifth post in this posts series.
The above is a basic example. It often isn’t as algebraically intuitive to calculate the detachment based on basic limit laws. In these cases, we may resort to the simple result from Semi-discrete Calculus (assuming $x$ is differentiable enough times and detachable):
$$x_{\pm}^{;}\left(t\right)=\left(\pm1\right)^{k+1}sgn\left[x^{\left(k\right)}\left(t\right)\right],\label{trend_based_on_rates}\tag{1}$$
where $k$ is the order of the first non-zeroed derivative of $x.$ This formula encapsulates the slightly more involved higher-order derivative tests.
Thus, we can think of the detachment as an auxiliary thought tool in trends calculations. We may apply it to calculate the trend directly by the definition if the limits algebra permits. Otherwise, assuming differentiability enough times and detachability, we may calculate the trend concisely with formula ($\ref{trend_based_on_rates}$). Depending on the context, some may opt to work with a single memorable formula rather than applying the conditions of the higher-order derivative test.
3.2 Charectarizing the stability of equilibriums in phase diagrams
In Physical Chemistry, Engineering, Mineralogy, and Materials science, we often use phase diagrams. It is a type of chart that shows conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous states) occur and coexist at equilibrium. Let us consider the dynamic model defined by $\left(x\left(t\right),y\left(t\right)\right),$ where $t\in\mathbb{R}^{+}$ is the time parameter. To find its equilibrium points, we solve the system of equations:
$$\begin{cases}
\dot{x}\left(t\right)=0,\\
\dot{y}\left(t\right)=0.
\end{cases}$$
Then, to characterize the equilibrium stability at each of the solutions, we calculate the derivative signs at each of the points’ environments. That is, while we care about the equilibrium at the point itself, we resort to calculations around it – due to the technical and mathematical contraint of the vanishing derivative there. Put differently, since the derivatives at the equilibriums inherently vanish, the derivative signs there don’t provide enough information as to the equilibrium type, and we have to think of the equilibrium as a function of the derivative sign at the points’ environment. Can we slightly simplify this process?
The (one-sided) detachments of $x,y$ always capture the functions’ trends at the equilibrium points (as long as the trends exist). And the trend is all we need to classify the equilibriums. Thus, depending on the context, some may opt to settle with calculating the detachments at the points themselves rather than the derivatives’ signs around them.
4 Continuous non-differentiable functions
4.1 Continuous functions with isolated singularities
In statistics, a power law is a functional relationship between two quantities. A relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. More than a hundred power-law distributions exist in Physics and Biology. While we could potentially reformulate any power rule as a fractional exponent by simply changing the formula’s subject, some of them naturally involve fractional exponents. Let us mention a few examples:
- In Astronomy:
- Kepler’s third law, that can be naturally rewritten with a fractional exponent if we isolate $T$ or $r.$
- The initial mass function of stars involves many different fractional exponents.
- The $M-\sigma$ relation
- In Physics:
- The Angstrom exponent in aerosol optics
- The frequency-dependency of acoustic attenuation in complex media
- Stevens’s power law of psychophysics
- Behaviour near second-order phase transitions involving critical exponents
- Curie–von Schweidler law
- In Biology:
- Kleiber’s law
- Taylor’s law
- In Astronomy:
The exponent is a fractional number in each of these laws, often rendering the power-law non-differentiable at a cusp point. Nevertheless, the function is detachable there. Thus, applying the detachment in that end case where the derivative is undefined complements the derivative’s work by enabling us to analyze the functions’ monotonicity everywhere.
4.2 Everywhere continuous nowhere differentiable functions
Physics is abundant with phenomena described by everywhere continuous and nowhere differentiable functions. A prominent example is fluctuations, random invisible movements of objects in their seemingly steady-state. These are studied in the fields of Thermodynamics and Quantum Mechanics, to mention a few. Another prominent example is the Brownian motion, whose statistical model – Wiener Process – resembles fluctuations.
Researchers have been suggesting several possible characterizations of this process, and we’ll specify one of them. Its value $W(t)$ adheres to the following:
- $W\left(0\right)=0$
- $W$ has independent increments: for every $t>0,$ the future increments $W\left(t+u\right)-W\left(t\right),$ $u\geq0$ are independent of the past values $W\left(s\right), s\leq t.$
- $W$ has Gaussian increments: $W\left(t+u\right)-W\left(t\right)$ is normally distributed with mean $0$ and variance $u$, $W\left(t+u\right)-W\left(t\right)\sim\mathcal{N}\left(0,u\right).$
- $W$ has continuous paths: $W\left(t\right)$ is continuous in $t.$
This charectarization of the Wiener process results in the following qualitative properties, as stated here:
- For every $\epsilon>0,$ the function $W$ takes both (strictly) positive and (strictly) negative values on $\left(0,\epsilon\right).$
- The function $W$ is continuous everywhere but differentiable nowhere.
- Points of local maximum of the function $W$ are a dense countable set.
- The function $W$ has no points of local increase, that is, no $t>0$ satisfies the following for some $\epsilon$ in $\left(0,t\right)$: first, $W\left(s\right)\leq W\left(t\right)$ for all $s$ in $\left(t-\epsilon,t\right),$ and, second, $W\left(s\right)\geq W\left(t\right)$ for all $s$ in $\left(t,t+\epsilon\right).$ (Local increase is a weaker condition than $W$ is increasing on $\left(t-\epsilon,t+\epsilon\right)).$ The same holds for local decrease.
Note how we can reformulate the above conditions in terms of the detachment operator:
This description is more concise and elegant than the former. It also states the existence of a property (detachability) at a dense subset (local optima), rather than the absence of differentiability everywhere.
5 Discontinuous functions
In Thermodynamics, a standard definition of “phase” is a state of matter whose properties vary smoothly (i.e., it is an analytic function of the pressure, volume, temperature, etc).
The definition itself implies that phase transitions involve discontinuities.
For example, this definition may mistakenly lead one to think that water and steam in the same phase, since we can boil water and it slowly becomes steam. However, this process is not smooth. For example, consider the temperature of water as heat is added. As we heat it the temperature rises. But when it hits the boiling point, the temperature does not rise anymore, instead the heat goes into vaporizing the water. Then onces it’s all gas, its temperature changes again. Hence, the density of $H_{2}O$, $\rho\left(T\right)$ changes discontinuously and non-analytically as a function of temperature around $T=100^{\circ}C .$
Another example is the first derivative of the Gibbs free energy, $G$, with respect to temperature. It is discontinuous across a phase boundary: infinitesimally below it is $\left(\frac{\partial G}{\partial T}\right)_{P}=-S_{\text{liquid}},$ and infinitesimally above it is $\left(\frac{\partial G}{\partial T}\right)_{P}=-S_{\text{gas}}.$ In a pure phase, $G$ is a smooth function (it and all its derivatives are continuous). Therefore, $G$ changes non-smoothly, at the phase changes.
More generally, a first-order phase transition is one for whom $\left(\frac{\partial G}{\partial T}\right)_{P}$ changes discontinuously at the phase boundary. It is possible for $\left(\frac{\partial G}{\partial T}\right)_{P}$ to be continuous, but higher derivatives of $G$ to be discontinuous. An $n^{\text{th}}$ order phase transition is on for whom the higher-order derivative, $\left(\frac{\partial^{n}G}{\partial T^{n}}\right)_{P}$ changes discontinuously at the phase boundary. This classification of phase transitions is known as the Ehrenfest classification. The control parameter whose continuity we are questioning in the transition is called the order parameter. For example, in the Ehrenfest classification, the entropy $S=-\left(\frac{\partial G}{\partial T}\right)_{P}$ is the order parameter.
An interesting phenomenon takes place at liquid-vapor critical points, the endpoints of the phase equilibrium curve. It isn’t easy to show that all of the derivatives of the pressure $P$ and the temperature $T$ vanish, $$\left(\frac{\partial^{n}P}{\partial v^{n}}\right)_{T}=\left(\frac{\partial^{n}T}{\partial v^{n}}\right)_{P}=0.$$ Thus, $P(v)$ and $T\left(v\right)$ are non-analytic functions at the critical point. Second-order phase transitions are interesting because this anomalous mathematical behavior, where all the derivatives vanish, arises out of functions like the entropy $S$ or the partition function $Z$ that depend smoothly on temperature, pressure, volume, etc.
It means that all the dimensionful physical quantities we use to characterize a material either vanish or are infinite at this point. Therefore, there are often discontinuities in the heat capacity, and other control parameters, at critical points.
To summarize this section, phase transitions are characterized with discontinuities of many types, of functions and their derivatives, across phase boundaries and particularly at critical points. While the derivative isn’t well defined there, the detachment is well defined. This helps us better explain and describe the phenomenon analytically. For example, if the heat capacity $C$ satisfies at a critical point $c$ with temperature $T_{c}$ that:
$$\underset{T\to T_{c}^{+}}{\lim}C\left(T\right)=\infty,\,\,C\left(T_{c}\right)<\infty,$$
then its detachment is properly defined,
$$C_{+}^{;}\left(T_{c}\right)=+1.$$
This helps us analyze $C$ continuously and concisely in spite of its discontinuity.
6 Summary
The derivative is a natural way to measure functions’ “instantaneous” rate of change. Mathematicians developed it in an era where most, if not all the known physical phenomena were smooth and continuous. However, Physics, as we know it today, is often non-differentiable. Further, the rigor of Mathematics has improved since the $17^{th}$ century. Calculus, as we know it today, allows us to distinguish between the detachment and the derivative sign with more solid notions of the limit process and continuity. The rise of computers also leads us to pay attention to the numerical difference between the detachment and the derivative sign. A function’s detachment describes its trend more coherently than the derivative sign in cases where the derivative either vanishes or doesn’t exist. It is also more numerically efficient, sparing up to 30% runtime in the computerized calculation of trends when compared with the derivative sign.
References
[1] Lovasz, L., 1998. One mathematics. Mitteilungen der Deutschen Mathematiker-Vereinigung, 6(2), pp.33-39.
[2] Schwartz, M., Spring 2019, Lecture 9: Phase Transitions, Statistical Mechanics, Harvard University.
[3] Strauss, D.F., 2017. Continuity and discontinuity in physics and biology. Suid-Afrikaans Tydskrif vir Natuurwetenskap en Tegnologie/South African Journal of Science and Technology, 36(1), pp.9-bladsye.
[4] Tao, T., 2013. Ultraproducts as a Bridge Between Discrete and Continuous Analysis. https://terrytao.wordpress.com/2013/12/07/ultraproducts-as-a-bridge-between-discrete-and-continuous-analysis/ “What’s new” – Terence Tao’s blog. Available at: https://terrytao.wordpress.com/2013/12/07/ultraproducts-as-a-bridge-between-discrete-and-continuous-analysis/
[5] 3Blue1Brown. (2017, April 29). The paradox of the derivative | Chapter 2, Essence of calculus [Video]. YouTube. https://www.youtube.com/watch?v=9vKqVkMQHKk