{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Emergence from chaos\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/emergence-from-chaos). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "A gas contains an enormous number of molecules. Newton's laws can describe each\n", "classical collision, yet no practical calculation follows every molecular position and\n", "velocity in a room. Thermodynamics succeeds by asking a different kind of question:\n", "which large-scale quantities remain reproducible when microscopic details vary?\n", "\n", "Pressure, temperature, density, and internal energy are macroscopic observables. They\n", "compress a vast microscopic state into a few numbers. That compression loses information,\n", "but it reveals regularities that are difficult to see in any single trajectory.\n", "\n", "## Deterministic does not mean practically predictable\n", "\n", "A deterministic model assigns one future to one exact initial state. It does not promise\n", "that the future can be predicted accurately when the initial state is known only to\n", "finite precision. In a chaotic system, nearby states can separate rapidly.\n", "\n", "During an interval in which the separation is still small, a common local description is\n", "\n", "\n", "$$\n", "\\delta(t)\\approx\\delta_0e^{\\lambda t}\n", "$$\n", "\n", "\n", "Here $\\delta_0$ is the initial\n", "separation, $\\delta(t)$ is the\n", "later separation, and a positive\n", "$\\lambda$ represents\n", "sensitivity to initial conditions. The relation is local: separation cannot grow\n", "exponentially forever in a bounded region, so it eventually saturates at the size of the\n", "accessible state space.\n", "\n", "Chaos is not the same as randomness. A chaotic classical trajectory may obey deterministic\n", "equations exactly. Randomness can instead enter through uncertain initial conditions,\n", "thermal contact, measurement noise, or a deliberately stochastic model. A dilute ideal\n", "gas with noninteracting particles is not automatically a chaotic dynamical system merely\n", "because the molecular velocities are described statistically. These distinctions prevent\n", "the word \"chaos\" from becoming a vague synonym for complexity.\n", "\n", "## Stable observables from unstable details\n", "\n", "Imagine two boxes prepared with the same number of gas molecules, volume, and total\n", "energy, but with slightly different molecular coordinates. Their later collision\n", "sequences can be entirely different. Nevertheless, the time-averaged pressure can be\n", "nearly the same because it depends on the accumulated momentum delivered to the walls,\n", "not on the identity of the molecule responsible for each collision.\n", "\n", "This is emergence: a robust large-scale pattern follows from many small-scale events even\n", "though the pattern is not a property of one event. No molecule carries \"the pressure\" of\n", "the gas. Pressure belongs to the collective behavior of molecules and the chosen system\n", "boundary.\n", "\n", "For sampled values $A_i$ of a\n", "macroscopic observable, the sample mean is\n", "\n", "\n", "$$\n", "\\bar A=\\frac{1}{M}\\sum_{i=1}^{M}A_i\n", "$$\n", "\n", "\n", "The observable must be defined before averaging. Pressure might mean wall impulse divided\n", "by wall area and elapsed time. Temperature might be inferred from random kinetic energy\n", "after subtracting the center-of-mass motion. Mixing incompatible definitions can produce\n", "a smooth number with no clear physical meaning.\n", "\n", "## Why relative fluctuations often become small\n", "\n", "Suppose a macroscopic total is the sum of many nearly independent contributions with\n", "similar finite variance. The mean total grows in proportion to the number of\n", "contributions, while its standard deviation grows roughly as the square root of that\n", "number. Therefore the fractional fluctuation scales approximately as\n", "\n", "\n", "$$\n", "\\frac{\\sigma_{\\mathrm{total}}}{\\langle A_{\\mathrm{total}}\\rangle}\n", " \\propto\\frac{1}{\\sqrt{N}}\n", "$$\n", "\n", "\n", "For a mole of matter, $N$ is so large\n", "that many ordinary relative fluctuations are tiny. The microscopic motion remains\n", "vigorous; it is the fraction represented by the fluctuation that becomes small.\n", "\n", "The inverse-square-root rule is not universal. Strong correlations make groups of\n", "particles move together and reduce the number of effectively independent contributions.\n", "Near a critical point, correlations can extend over large distances and fluctuations can\n", "become conspicuous. Very small systems also fluctuate substantially. A reliable\n", "explanation therefore states the independence or short-correlation assumption rather\n", "than quoting the scaling law without conditions.\n", "\n", "## Ensemble averages and time averages\n", "\n", "An ensemble average compares many systems prepared according to the same macroscopic\n", "conditions at one selected time. A time average follows one system over a selected\n", "interval. These procedures answer different experimental questions.\n", "\n", "Ergodic reasoning says that, for suitable equilibrium systems and observables, a\n", "sufficiently long time average is representative of the corresponding ensemble average.\n", "Ergodicity is not guaranteed by deterministic dynamics, chaos, or a long run alone. A\n", "system can remain trapped in one region of state space, retain a conserved quantity that\n", "was overlooked, or relax on a time scale longer than the observation.\n", "\n", "Successive measurements from one trajectory are also correlated. Recording twice as\n", "often does not necessarily double the information. If correlations persist across many\n", "samples, the effective sample count is smaller than the stored sample count, and an\n", "independent-sample error bar is overconfident.\n", "\n", "The distinction can be organized without a wide expression:" ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "\n", "def logistic_samples(start, count=12000):\n", " x, values = start, []\n", " for step in range(count):\n", " x = 4 * x * (1 - x)\n", " if step >= 2000:\n", " values.append(x)\n", " return np.array(values)\n", "\n", "\n", "for start in [0.123, 0.321, 0.456]:\n", " values = logistic_samples(start)\n", " print(start, \"mean:\", values.mean(), \"variance:\", values.var())\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "This procedure does not force equilibrium. It exposes whether different preparations\n", "approach the same macroscopic behavior within the stated uncertainty.\n", "\n", "## Relaxation, mixing, and equilibrium\n", "\n", "Collisions redistribute energy and momentum among molecules. A localized excess of\n", "particles can spread, and an initially anisotropic velocity distribution can become more\n", "nearly isotropic. The macroscopic state then relaxes toward equilibrium even though each\n", "microscopic collision preserves momentum and energy for the isolated colliding pair.\n", "\n", "Equilibrium does not mean that every molecule stops changing. It means that selected\n", "macroscopic observables have stationary probability distributions and no sustained\n", "macroscopic gradients or flows remain inside the stated boundary. Instantaneous values\n", "continue to fluctuate.\n", "\n", "Different observables can relax at different rates. A mean kinetic energy may settle\n", "before a spatial density imbalance disappears. Declaring equilibrium from one flat curve\n", "can therefore be premature. Check several physically relevant observables and compare\n", "multiple time windows.\n", "\n", "## Reversibility and the macroscopic arrow\n", "\n", "Classical collision laws are approximately reversible: reverse every molecular momentum\n", "in an idealized isolated system and the equations retrace the former motion. Macroscopic\n", "irreversibility does not arise because those equations suddenly change direction. It\n", "arises statistically because equilibrium-like macrostates correspond to overwhelmingly\n", "more microscopic arrangements than specially organized nonequilibrium macrostates.\n", "\n", "Boltzmann's relation expresses this connection:\n", "\n", "\n", "$$\n", "S=k_B\\ln\\Omega\n", "$$\n", "\n", "\n", "The symbol $\\Omega$ counts\n", "the microscopic states compatible with a specified macrostate, and\n", "$S$ is its statistical entropy. A system prepared\n", "in a low-multiplicity macrostate is overwhelmingly likely to move toward a\n", "higher-multiplicity macrostate because there are far more compatible ways to be there.\n", "\n", "This is a probabilistic statement for macroscopic systems, not a claim that a fluctuation\n", "toward lower entropy is mathematically impossible. In a small system, temporary reverse\n", "fluctuations can be observable. In a macroscopic system, a large spontaneous reversal is\n", "so improbable that thermodynamics treats it as absent on practical time scales.\n", "\n", "## Physical divergence versus numerical error\n", "\n", "A numerical trajectory can diverge from another for two different reasons. The physical\n", "model may be chaotic, amplifying a tiny initial difference, or the numerical method may\n", "introduce truncation and roundoff errors. Usually both effects are present.\n", "\n", "Trajectory disagreement after a long time is therefore not, by itself, evidence that an\n", "implementation is wrong. Conversely, the label \"chaos\" must not excuse systematic energy\n", "drift, incorrect collision impulses, biased boundary handling, or dependence on the time\n", "step.\n", "\n", "Test reproducibility at two levels:" ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "first = logistic_samples(0.123)\n", "replay = logistic_samples(0.123)\n", "perturbed = logistic_samples(0.123 + 1e-10)\n", "assert np.array_equal(first, replay)\n", "print(\"Trajectory difference:\", np.mean(abs(first - perturbed)))\n", "print(\"Difference of long-window means:\", abs(first.mean() - perturbed.mean()))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "The microscopic traces may lose pointwise agreement while distributions and macroscopic\n", "averages converge. That is the expected signature of stable emergent behavior. If the\n", "averages shift systematically as the numerical step is reduced, the calculation has not\n", "yet reached numerical convergence.\n", "\n", "## Evidence for an emergent claim\n", "\n", "A credible macroscopic claim names the system boundary, observable, units, preparation\n", "procedure, equilibration interval, sampling interval, and averaging window. It shows the\n", "raw variation as well as the aggregate. It also reports how uncertainty was estimated and\n", "whether temporal correlation was considered.\n", "\n", "Compare multiple initial preparations. Repeat the calculation with a smaller numerical\n", "step. Check conservation laws that apply to the chosen boundary. Increase the number of\n", "particles when feasible and ask whether relative fluctuations shrink in the predicted\n", "direction. Finally, change the observation window and ensure the conclusion is not an\n", "artifact of a convenient start or stop time.\n", "\n", "The essential lesson is not that disorder mysteriously creates order. The microscopic\n", "description contains more detail than a macroscopic question needs. Averaging, large\n", "numbers, conservation laws, and the relative multiplicities of macrostates make some\n", "collective quantities far more stable than individual trajectories.\n", "\n", "## A worked Python notebook\n", "\n", "Examine the deterministic logistic update used to compare microscopic divergence and long-run distributions. The question for this calculation is precise: how does **state after one iteration**\n", "change when we change **initial logistic-map state**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "One iteration is not a thermodynamic model; it isolates a simple nonlinear mapping. Keep that sentence beside the results. A formula can return a number outside\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "0 to 1 dimensionless. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for initial logistic-map state, expressed in dimensionless. The output is\n", "state after one iteration, expressed in dimensionless. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = 4 q \\left(1 - q\\right)\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 0.3 dimensionless, the expression gives 0.84 dimensionless. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"4*q*(1-q)\", locals={\"q\": q})\n", "reference_input = 0.3\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"dimensionless\")\n", "print(\"Predicted state after one iteration:\", reference_output, \"dimensionless\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "0 dimensionless. The midpoint, at 0.5 dimensionless, gives\n", "1 dimensionless. The upper endpoint gives 0 dimensionless.\n", "Read these together with the assumptions: they describe different choices of initial logistic-map state\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.0, 0.5, 1.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} dimensionless -> {output_value:.6g} dimensionless\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows initial logistic-map state; the vertical axis shows state after one iteration. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "input_grid = np.linspace(0.0, 1.0, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Initial logistic-map state (dimensionless)\")\n", "ax.set_ylabel(\"State after one iteration (dimensionless)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "state after one iteration change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.3 dimensionless is\n", "1.6 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = 4 - 8 q\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.305\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"dimensionless\")\n", "print(\"Full expression:\", full_prediction, \"dimensionless\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"dimensionless\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Here the changed input is 0.305 dimensionless. The tangent estimate is\n", "0.848 dimensionless, while the full expression gives\n", "0.8479 dimensionless. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_span = 1.0\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "initial logistic-map state inside the stated interval, predict whether state after one iteration will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: One iteration is not a thermodynamic model; it isolates a simple nonlinear mapping. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of dimensionless:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(1.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 1.0, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 2*, Chapter 2, “The Kinetic Theory of Gases”](https://openstax.org/books/university-physics-volume-2/pages/2-introduction).\n", "- [OpenStax, *University Physics Volume 2*, Section 4.7, “Entropy on a Microscopic Scale”](https://openstax.org/books/university-physics-volume-2/pages/4-7-entropy-on-a-microscopic-scale).\n", "- [MIT OpenCourseWare 8.01SC, Chapter 29, “Kinetic Theory of Gases”](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter29/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 16, “Thermodynamics”](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, “Kinetic-Molecular Theory”](https://physics.info/kmt/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "emergence-from-chaos.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/emergence-from-chaos", "source_sha256": "c1beb3ccfed6d4096f1b53f3ebaf90aa550aaf98ffd2ec42e478e82c4c15ff46" } }, "nbformat": 4, "nbformat_minor": 5 }