{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Irreversibility in simulation\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/irreversibility-in-simulation). 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": [ "Every numerical model advances from an earlier state to a later state, but forward\n", "execution alone is not a physical arrow of time. A model can fail to return to its initial\n", "state because the physical process is thermodynamically irreversible, because the\n", "microscopic trajectory is chaotic, because the numerical update has no exact inverse, or\n", "because information was discarded. Those causes must be separated.\n", "\n", "## Three meanings of reversible\n", "\n", "Microscopic time-reversal symmetry concerns equations of motion. For an isolated\n", "classical system with position-dependent conservative forces, reversing all momenta maps\n", "a valid trajectory into one that retraces the motion in reverse time.\n", "\n", "Numerical reversibility concerns an update algorithm. If a discrete update maps state\n", "$x_n$ to\n", "$x_{n+1}$, it is reversible only\n", "if the previous state can be recovered under the declared arithmetic and inverse rule.\n", "A good approximation to reversible differential equations need not be exactly invertible\n", "after floating-point rounding.\n", "\n", "Thermodynamic reversibility is stricter. A reversible thermodynamic process can restore\n", "both the system and its surroundings to their initial states through an infinitesimal\n", "change in external conditions. It must be quasistatic and free from dissipative effects.\n", "A quasistatic process with friction is still irreversible.\n", "\n", "Confusing these definitions creates false conclusions. A time-symmetric particle model\n", "does not make a rapidly expanding gas thermodynamically reversible, and an irreversible\n", "computer operation does not prove that a physical system produced thermodynamic entropy.\n", "\n", "## Entropy production needs a boundary\n", "\n", "For a process involving a system and its surroundings, the second law can be written\n", "\n", "\n", "$$\n", "\\Delta S_{\\mathrm{universe}}\n", " =\\Delta S_{\\mathrm{system}}\n", " +\\Delta S_{\\mathrm{surroundings}}\\ge0\n", "$$\n", "\n", "\n", "The equality applies to an ideal reversible process. A macroscopic irreversible process\n", "has a positive total change. The system entropy alone can decrease if a larger entropy\n", "increase occurs in the surroundings, as in a refrigerator. Therefore a calculation that\n", "tracks only visible particles but omits a thermostat or frictional reservoir cannot\n", "establish the total entropy balance.\n", "\n", "Entropy is a state function. For a reversible path between equilibrium states,\n", "\n", "\n", "$$\n", "\\Delta S=\\int_i^f\\frac{\\delta Q_{\\mathrm{rev}}}{T}\n", "$$\n", "\n", "\n", "The subscript on heat is essential. For an actual irreversible path, dividing the actual\n", "heat transfer by temperature does not generally produce the system's entropy change.\n", "Instead, choose a hypothetical reversible path connecting the same equilibrium endpoints\n", "or use an appropriate state equation.\n", "\n", "## Free expansion exposes the distinction\n", "\n", "Consider an ideal gas initially confined to part of a rigid insulated container. Removing\n", "a partition lets it expand into a vacuum. The gas does no boundary work, receives no\n", "heat, and has no change in internal energy:\n", "\n", "\n", "$$\n", "Q=0,\\qquad W=0,\\qquad\\Delta U=0\n", "$$\n", "\n", "\n", "For an ideal gas, unchanged internal energy means unchanged temperature. Yet the entropy\n", "increases:\n", "\n", "\n", "$$\n", "\\Delta S=nR\\ln\\left(\\frac{V_f}{V_i}\\right)>0\n", "$$\n", "\n", "\n", "Here $n$ is the amount of gas,\n", "$R$ is the gas constant, and\n", "$V_f>V_i$ gives\n", "the positive sign. The entropy change is evaluated using a reversible isothermal path\n", "between the same endpoints, not by substituting the actual zero heat into the reversible\n", "formula.\n", "\n", "A particle calculation may preserve every microscopic collision and still show the gas\n", "spreading. If every momentum were reversed with impossible precision, the microscopic\n", "trajectory could reconcentrate. The reversed state is extraordinarily special among the\n", "states compatible with the expanded macrostate, so ordinary uncontrolled preparations do\n", "not produce it.\n", "\n", "## Fine-grained and coarse-grained descriptions\n", "\n", "Hamiltonian dynamics preserves phase-space volume. Under ideal exact evolution, the\n", "fine-grained Gibbs entropy of a complete probability density remains constant. The\n", "density can stretch and fold into structures too fine for a practical observer to\n", "resolve.\n", "\n", "Coarse-graining groups many nearby microstates into cells or histogram bins. As the\n", "fine-scale structure becomes unresolved, the coarse-grained distribution can spread over\n", "more cells and its entropy can rise. This describes lost resolution in the macroscopic\n", "description, not destruction of the exact microscopic information by the Hamiltonian\n", "equations.\n", "\n", "A plotted histogram therefore needs its bin boundaries, sampling interval, and state\n", "variables. Changing the binning during a run can create an artificial entropy trend.\n", "Compare like with like.\n", "\n", "## Physical dissipation is energy transfer\n", "\n", "Friction, viscosity, inelastic deformation, and electrical resistance transform organized\n", "macroscopic energy into less organized internal energy. A physical model should conserve\n", "total energy when all relevant reservoirs are included, even though the visible\n", "mechanical energy decreases.\n", "\n", "Deleting a fraction of velocity at every step creates damping, but it does not explain\n", "where the removed energy goes. A better accounting records the lost mechanical energy as\n", "energy gained by an internal reservoir or an external bath. Momentum exchange also needs\n", "a destination when the boundary exerts an impulse.\n", "\n", "For a damped mechanical subsystem,\n", "\n", "\n", "$$\n", "\\Delta E_{\\mathrm{mechanical}}\n", " +\\Delta E_{\\mathrm{internal}}\n", " +E_{\\mathrm{boundary}}=0\n", "$$\n", "\n", "\n", "The signs depend on the chosen boundary convention, so it must be stated. Entropy\n", "production additionally depends on temperatures and reservoir changes; energy loss from\n", "one displayed variable is not enough.\n", "\n", "## Numerical sources of irreversibility\n", "\n", "Finite-precision arithmetic maps many exact real numbers to one stored value. Rounding is\n", "many-to-one and cannot be inverted without extra information. Clipping, filtering,\n", "histogramming, image rendering, state compression, and deleting old checkpoints are also\n", "lossy.\n", "\n", "Common time integrators introduce truncation error. Some, such as properly implemented\n", "velocity Verlet for conservative forces, are time-symmetric, while ordinary forward\n", "Euler is not. Adaptive step selection can make reversal difficult because the backward\n", "run may choose a different step sequence. Floating-point operation order can also change\n", "rounding.\n", "\n", "Chaotic sensitivity amplifies these small differences. A failed long-time trajectory\n", "reversal may therefore indicate tiny numerical discrepancies amplified by the dynamics,\n", "not macroscopic entropy production. Conserved quantities and convergence of statistical\n", "observables are stronger diagnostics than visual retracing alone.\n", "\n", "## Random processes require more than a seed\n", "\n", "A stochastic update consumes random values. Reusing the same initial seed reproduces a\n", "forward stream only if the generator algorithm, state, draw order, and implementation are\n", "unchanged. It does not automatically supply the random choices needed for the inverse\n", "transition.\n", "\n", "If the transition is many-to-one, no seed can reconstruct information that the update\n", "discarded. Exact replay needs checkpoints or a log of choices. Statistical reversibility,\n", "such as detailed balance in an equilibrium Markov chain, is a statement about transition\n", "probabilities and is different from reversing one sampled path.\n", "\n", "## A layered reverse test\n", "\n", "Start with the smallest deterministic case. Test one forward update followed by its\n", "declared inverse before attempting a long chaotic trajectory." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "\n", "def verlet(state, dt):\n", " x, v = state\n", " new_x = x + dt * v - 0.5 * dt**2 * x\n", " new_v = v - 0.5 * dt * (x + new_x)\n", " return np.array([new_x, new_v])\n", "\n", "\n", "initial = np.array([1.0, 0.2])\n", "state = initial.copy()\n", "for _ in range(1000):\n", " state = verlet(state, 0.01)\n", "state[1] *= -1\n", "for _ in range(1000):\n", " state = verlet(state, 0.01)\n", "state[1] *= -1\n", "print(\"Reversal error:\", np.linalg.norm(state - initial))\n", "assert np.allclose(state, initial)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Then repeat while adding one potentially irreversible element at a time: rounding,\n", "damping, a thermostat, a stochastic collision, or coarse-grained storage. This isolates\n", "the first loss rather than attributing every discrepancy to \"entropy.\"\n", "\n", "Use a dimensionless recovery measure only after defining physical scales. One option is\n", "\n", "\n", "$$\n", "\\epsilon_{\\mathrm{rec}}\n", " =\\frac{\\lVert x_{\\mathrm{recovered}}-x_{\\mathrm{initial}}\\rVert}\n", " {x_{\\mathrm{scale}}}\n", "$$\n", "\n", "\n", "State which components enter the norm and how positions, momenta, and other quantities\n", "are nondimensionalized. Adding meters directly to momentum units would make the norm\n", "meaningless.\n", "\n", "## Evidence for a physical conclusion\n", "\n", "A strong analysis reports the system boundary, initial macrostate, microscopic dynamics,\n", "integrator, numerical step, precision, reversal rule, and every modeled reservoir. It\n", "checks energy and momentum where they should be conserved and includes the surroundings\n", "in the entropy balance.\n", "\n", "It also distinguishes three outcomes. Exact state recovery tests the discrete algorithm.\n", "Convergence under smaller steps tests numerical approximation. Stable distributions and\n", "macroscopic averages across independent preparations test the statistical prediction.\n", "Passing one test does not imply the others.\n", "\n", "The central lesson is that irreversibility is not a single symptom. Thermodynamic entropy\n", "production, coarse-grained information loss, physical dissipation, chaotic amplification,\n", "and non-invertible computation can coexist, but they are different claims requiring\n", "different evidence.\n", "\n", "## A worked Python notebook\n", "\n", "Use a reversible harmonic trajectory as a reference for a forward-and-reverse computation. The question for this calculation is precise: how does **ideal oscillator position**\n", "change when we change **time**, 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", "The equation omits damping; failure to recover the initial state can then reveal numerical information loss. 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 6 s. 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 time, expressed in s. The output is\n", "ideal oscillator position, expressed in m. 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", "$$\n", "f(q) = 0.2 \\sin{\\left(q \\right)} + \\cos{\\left(q \\right)}\n", "$$\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 2 s, the expression gives -0.234287 m. 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(\"cos(q)+0.2*sin(q)\", locals={\"q\": q})\n", "reference_input = 2.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"s\")\n", "print(\"Predicted ideal oscillator position:\", reference_output, \"m\")\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", "1 m. The midpoint, at 3 s, gives\n", "-0.961768 m. The upper endpoint gives 0.904287 m.\n", "Read these together with the assumptions: they describe different choices of time\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([0.0, 3.0, 6.0])\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} s -> {output_value:.6g} m\")\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 time; the vertical axis shows ideal oscillator position. 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(0.0, 6.0, 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(\"Time (s)\")\n", "ax.set_ylabel(\"Ideal oscillator position (m)\")\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", "ideal oscillator position change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 2 s is\n", "-0.992527 in those ratio units. Locally, the output decreases 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", "$$\n", "f'(q) = - \\sin{\\left(q \\right)} + 0.2 \\cos{\\left(q \\right)}\n", "$$\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 = 2.03\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, \"m\")\n", "print(\"Full expression:\", full_prediction, \"m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Here the changed input is 2.03 s. The tangent estimate is\n", "-0.264063 m, while the full expression gives\n", "-0.263953 m. 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 = 6.0\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-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", "time inside the stated interval, predict whether ideal oscillator position 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: The equation omits damping; failure to recover the initial state can then reveal numerical information loss. 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 m:\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(6.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 6.0, 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-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 3.4, “Thermodynamic Processes”](https://openstax.org/books/university-physics-volume-2/pages/3-4-thermodynamic-processes).\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”](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)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "irreversibility-in-simulation.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/irreversibility-in-simulation", "source_sha256": "bce5280c31edbb001203634e36cb9ddbe816e6092789998bdf0a51808111bddd" } }, "nbformat": 4, "nbformat_minor": 5 }