{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Entropy as information\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/entropy-as-information). 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": [ "Entropy appears in both information theory and statistical physics. The formulas are\n", "closely related because both fields quantify uncertainty over possible states. The\n", "interpretations and units are not automatically interchangeable, however. An information\n", "entropy belongs to a declared probability distribution and logarithm base; a\n", "thermodynamic entropy also depends on the physical ensemble and carries units of energy\n", "per temperature.\n", "\n", "## Information carried by an outcome\n", "\n", "An outcome that was nearly certain provides little new information when observed. A rare\n", "outcome provides more. For an outcome with probability\n", "$p_i$, its self-information in\n", "base $b$ is\n", "\n", "\n", "$$\n", "I_i=-\\log_b p_i\n", "$$\n", "\n", "\n", "The minus sign makes the information nonnegative because probabilities do not exceed\n", "one. An event with probability one has zero self-information. As the probability becomes\n", "smaller, the information grows.\n", "\n", "Shannon entropy is the expected self-information:\n", "\n", "\n", "$$\n", "H=-\\sum_i p_i\\log_b p_i\n", "$$\n", "\n", "\n", "In plain language, multiply the information associated with each possible outcome by how\n", "often that outcome is expected, then add the contributions.\n", "\n", "When the logarithm has base two, entropy is measured in bits. Natural logarithms produce\n", "nats. Changing the base rescales the numerical value:\n", "\n", "\n", "$$\n", "\\log_b p=\\frac{\\ln p}{\\ln b}\n", "$$\n", "\n", "\n", "A numerical result must therefore name its base. Saying only \"the entropy is one\" is\n", "incomplete.\n", "\n", "## Certain and uniform distributions\n", "\n", "A distribution with one certain outcome has zero entropy. For two outcomes with\n", "probabilities one and zero,\n", "\n", "\n", "$$\n", "H(1,0)=0\n", "$$\n", "\n", "\n", "The term containing zero probability is defined by the limit\n", "\n", "\n", "$$\n", "\\lim_{p\\to0^+}p\\log p=0\n", "$$\n", "\n", "\n", "Code should handle that limit explicitly. Evaluating the logarithm of zero first produces\n", "an invalid intermediate value even though the complete limiting contribution is zero.\n", "\n", "For a fixed set of $K$\n", "outcomes, entropy is largest when every outcome is equally likely:\n", "\n", "\n", "$$\n", "p_i=\\frac{1}{K},\n", " \\qquad H_{\\max}=\\log_b K\n", "$$\n", "\n", "\n", "Two equally likely outcomes have one bit of entropy. Four equally likely outcomes have\n", "two bits. Uniformity maximizes uncertainty only after the allowed outcome set has been\n", "fixed; adding impossible labels to the data structure must not enlarge the physical\n", "state space.\n", "\n", "## From Shannon entropy to statistical entropy\n", "\n", "Statistical mechanics often uses the Gibbs entropy\n", "\n", "\n", "$$\n", "S=-k_B\\sum_i p_i\\ln p_i\n", "$$\n", "\n", "\n", "Here $S$ has SI units\n", "of joules per kelvin because it is multiplied by Boltzmann's constant\n", "$k_B$. The natural logarithm is\n", "conventional. The probabilities refer to physical microstates in a specified ensemble,\n", "not arbitrary message symbols.\n", "\n", "If $\\Omega$ microstates\n", "are equally likely, every probability is\n", "$1/\\Omega$. Substitution\n", "gives Boltzmann's relation:\n", "\n", "\n", "$$\n", "S=k_B\\ln\\Omega\n", "$$\n", "\n", "\n", "Thus Boltzmann entropy is a special equal-probability case of the Gibbs expression. The\n", "matching mathematical structure does not mean that the Shannon entropy of any computer\n", "file is the thermodynamic entropy of the device storing it. A physical mapping must\n", "identify the microstates, probabilities, constraints, and energy exchanges.\n", "\n", "## Macrostate information and hidden detail\n", "\n", "Let a random variable $X$\n", "identify a microstate, and let\n", "$M=f(X)$ be a deterministic\n", "macrostate label. The macrostate retains less detail than the microstate. Their Shannon\n", "entropies obey the chain relation\n", "\n", "\n", "$$\n", "H(X)=H(M)+H(X\\mid M)\n", "$$\n", "\n", "\n", "The conditional entropy\n", "$H(X\\mid M)$ measures\n", "the microscopic information still unknown after the macrostate is given. For an equally\n", "weighted macrostate with multiplicity\n", "$\\Omega(M)$, this hidden\n", "uncertainty is the logarithm of that multiplicity.\n", "\n", "This distinction prevents a common contradiction. Replacing a detailed state by a\n", "macrostate label does not create more Shannon information in the label. It discards\n", "detail. Thermodynamic entropy is related to how many microscopic possibilities remain\n", "compatible with the macroscopic information. A high-multiplicity macrostate leaves more\n", "microscopic uncertainty.\n", "\n", "## Coarse-graining changes the question\n", "\n", "Probabilities depend on how outcomes are grouped. A speed histogram with ten bins and one\n", "with one hundred bins describe the same measurements at different resolution, but their\n", "discrete entropies need not match. Merging bins loses distinctions; splitting bins\n", "introduces distinctions that require enough data to estimate reliably.\n", "\n", "For a continuous probability density, the differential entropy is\n", "\n", "\n", "$$\n", "h=-\\int p(x)\\ln p(x)\\,dx\n", "$$\n", "\n", "\n", "Unlike discrete entropy, differential entropy can be negative and changes under a\n", "rescaling of coordinates. Its numerical value cannot be interpreted as an ordinary count\n", "of uncertain bits without a reference measure or resolution. Relative entropy and\n", "physically defined entropy differences are often more robust than an isolated\n", "differential-entropy number.\n", "\n", "The coarse-graining rule must remain fixed when comparing times. Otherwise an apparent\n", "entropy change may come entirely from changing bins, units, or outcome labels.\n", "\n", "## Relative entropy compares distributions\n", "\n", "The Kullback-Leibler divergence measures how one probability distribution differs from a\n", "reference distribution:\n", "\n", "\n", "$$\n", "D_{\\mathrm{KL}}(p\\parallel q)\n", " =\\sum_i p_i\\ln\\frac{p_i}{q_i}\n", "$$\n", "\n", "\n", "It is nonnegative and equals zero only when the two distributions agree wherever the\n", "comparison is defined. It is not symmetric, so it is not a distance in the geometric\n", "sense. In statistical physics it can quantify departure from an equilibrium reference,\n", "provided the outcome space and support are shared.\n", "\n", "If the reference assigns zero probability where the measured distribution assigns a\n", "positive probability, the divergence is infinite. Silently adding a small replacement\n", "probability changes the model and must be reported.\n", "\n", "## Estimating entropy from data\n", "\n", "Measured data usually provide counts rather than exact probabilities. A plug-in estimate\n", "normalizes the counts and substitutes the observed frequencies into the entropy formula." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "counts = np.array([10, 20, 30, 40], dtype=float)\n", "if np.any(counts < 0) or counts.sum() <= 0:\n", " raise ValueError(\"Counts must be nonnegative and have a positive total.\")\n", "probabilities = counts[counts > 0] / counts.sum()\n", "entropy_bits = -np.sum(probabilities * np.log2(probabilities))\n", "print(f\"Shannon entropy: {entropy_bits:.6f} bits\")\n", "assert entropy_bits <= np.log2(len(counts))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "This estimator is generally biased downward for finite samples because unseen outcomes\n", "receive zero observed frequency. The bias becomes important when the number of possible\n", "outcomes is large compared with the sample count. Correlated samples reduce the effective\n", "information further. Bootstrap intervals, Bayesian estimators, or analytic corrections\n", "may help, but each introduces assumptions that belong in the report.\n", "\n", "## What entropy does and does not establish\n", "\n", "Information entropy quantifies uncertainty in a distribution. By itself it does not show\n", "that a physical process is irreversible, that energy was dissipated, or that a system is\n", "in thermal equilibrium. A shuffled list can have a different empirical symbol\n", "distribution without any thermodynamic claim, while a thermodynamic process can produce\n", "entropy in reservoirs that are absent from a narrow data record.\n", "\n", "To connect an information calculation to physics, name the physical state space,\n", "ensemble, conserved quantities, coarse-graining, and boundary. Explain why the\n", "probabilities represent the system and why the selected entropy is the relevant physical\n", "quantity.\n", "\n", "Useful checks are simple and strict. Probabilities must be nonnegative and sum to one.\n", "Reordering outcome labels must not change entropy. A certain outcome must give zero, and\n", "a uniform distribution over the same fixed support must give the maximum. Repeating an\n", "estimate with more data and alternative fixed bin widths reveals whether the conclusion\n", "is stable or resolution-dependent.\n", "\n", "## A worked Python notebook\n", "\n", "Compare uncertainty for two outcomes as their probabilities become more or less balanced. The question for this calculation is precise: how does **Shannon entropy**\n", "change when we change **probability of first binary outcome**, 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", "Shannon entropy in bits is not automatically thermodynamic entropy in joules per kelvin. 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", "0.01 to 0.99 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 probability of first binary outcome, expressed in dimensionless. The output is\n", "Shannon entropy, expressed in bits. 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", "$$\n", "f(q) = \\frac{- q \\log{\\left(q \\right)} - \\left(1 - q\\right) \\log{\\left(1 - q \\right)}}{\\log{\\left(2 \\right)}}\n", "$$\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 0.25 dimensionless, the expression gives 0.811278 bits. 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-005", "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(\"-(q*log(q)+(1-q)*log(1-q))/log(2)\", locals={\"q\": q})\n", "reference_input = 0.25\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"dimensionless\")\n", "print(\"Predicted Shannon entropy:\", reference_output, \"bits\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "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", "0.0807931 bits. The midpoint, at 0.5 dimensionless, gives\n", "1 bits. The upper endpoint gives 0.0807931 bits.\n", "Read these together with the assumptions: they describe different choices of probability of first binary outcome\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-007", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.01, 0.5, 0.99])\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} bits\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "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 probability of first binary outcome; the vertical axis shows Shannon entropy. 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-009", "metadata": {}, "source": [ "input_grid = np.linspace(0.01, 0.99, 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(\"Probability of first binary outcome (dimensionless)\")\n", "ax.set_ylabel(\"Shannon entropy (bits)\")\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-010", "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", "Shannon entropy change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.25 dimensionless is\n", "1.58496 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", "$$\n", "f'(q) = \\frac{- \\log{\\left(q \\right)} + \\log{\\left(1 - q \\right)}}{\\log{\\left(2 \\right)}}\n", "$$\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-011", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.2549\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, \"bits\")\n", "print(\"Full expression:\", full_prediction, \"bits\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"bits\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Here the changed input is 0.2549 dimensionless. The tangent estimate is\n", "0.819044 bits, while the full expression gives\n", "0.818952 bits. 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-013", "metadata": {}, "source": [ "input_span = 0.98\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-014", "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", "probability of first binary outcome inside the stated interval, predict whether Shannon entropy 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: Shannon entropy in bits is not automatically thermodynamic entropy in joules per kelvin. 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 bits:\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-015", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(0.99) - evaluate(0.01))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.01, 0.99, 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-016", "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*, 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", "- [Benjamin Crowell, *Light and Matter*, Chapter 16, “Thermodynamics,” and Chapter 33, “Rules of Randomness”](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [Physics LibreTexts, “Thermodynamics and Statistical Mechanics”](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics).\n", "- [Claude E. Shannon, “A Mathematical Theory of Communication”](https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "entropy-as-information.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/entropy-as-information", "source_sha256": "538c45d1fef6984248db97551e44bf4340f627aa9117ceca5a519b772684c229" } }, "nbformat": 4, "nbformat_minor": 5 }