.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/fields/gauge_confinement/plot_flux_tube_confinement.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_fields_gauge_confinement_plot_flux_tube_confinement.py: A confining flux tube stretching between two charges ========================================================== Kenneth Wilson's 1974 lattice gauge theory ("Confinement of Quarks") showed that, in the strong-coupling limit, the potential energy of a static quark-antiquark pair grows *linearly* with separation, rather than falling off the way an ordinary Coulomb field does -- the defining signature of confinement. The physical picture is that the gluon field lines are squeezed into a narrow "flux tube" of fixed cross-section connecting the two charges, rather than spreading out in all directions the way an ordinary Coulomb field would. :func:`~physicskit.fields.electrodynamics.flux_tube_field_1d` builds this toy model directly from the 1D Gauss law relating the confined flux :math:`\Phi(x)` to the charge density :math:`\rho(x)`, .. math:: \frac{d\Phi}{dx} = \rho(x), \qquad \rho(x) = Q\Big[g_w(x + s/2) - g_w(x - s/2)\Big], where the two opposite point charges of magnitude :math:`Q` (``flux_quantum``), placed a separation :math:`s` apart at :math:`x=\mp s/2`, are each smoothed into a Gaussian :math:`g_w` of width ``wall_width`` (a numerical regularization of the point charge, not a physical effect); :math:`\Phi(x)` is obtained by direct numerical integration of :math:`\rho`. :func:`~physicskit.fields.electrodynamics.flux_tube_energy_density_2d` extrudes this 1D flux into a 2D energy-density map by giving it a fixed transverse (``tube_width``) Gaussian profile -- the "fixed cross-section" that produces confinement -- .. math:: u(x, y) = \tfrac12\,\Phi(x)^2\, e^{-y^2/(2\,w_\perp^2)}. Because Gauss's law forces :math:`\Phi` to stay pinned at :math:`\pm Q` along the whole length of the tube regardless of how far apart the charges are, the total field energy :math:`\int u\,dx\,dy` grows *linearly* with the separation :math:`s` -- the confinement signature, reproduced here by direct construction rather than derived from an underlying gauge-invariant Lagrangian. :func:`~physicskit.fields.electrodynamics.flux_tube_field_1d` and :func:`~physicskit.fields.electrodynamics.flux_tube_energy_density_2d` reproduce only this qualitative signature, as a simplified 1+1D toy model -- **not** a lattice-QCD calculation. .. GENERATED FROM PYTHON SOURCE LINES 49-55 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.fields import animate_flux_tube, flux_tube_energy_density_2d .. GENERATED FROM PYTHON SOURCE LINES 56-63 Stretch the flux tube by animating a sweep of charge separations -------------------------------------------------------------------- The two charges live on a 200x40 grid and are swept from a separation of 4 to 16 grid units; at each separation, :func:`animate_flux_tube` re-solves the static field profile from scratch (there is no time dependence within a single separation -- the "animation" is over the quasi-static parameter :math:`s`, not real time). .. GENERATED FROM PYTHON SOURCE LINES 63-71 .. code-block:: Python shape = (200, 40) dx = dy = 0.2 separations = np.linspace(4.0, 16.0, 25) anim = animate_flux_tube(shape, dx, dy, separations) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 72-76 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("flux_tube.gif", writer="pillow", fps=12) .. GENERATED FROM PYTHON SOURCE LINES 78-81 The total confined field energy grows linearly with separation -- a straight line, not a Coulomb-like falloff or a plateau -- exactly the confinement signature Wilson's lattice calculation predicted. .. GENERATED FROM PYTHON SOURCE LINES 81-89 .. code-block:: Python total_energy = np.array([flux_tube_energy_density_2d(shape, dx, dy, sep).sum() * dx * dy for sep in separations]) fit = np.polyfit(separations, total_energy, 1) linfit_error = np.max(np.abs(total_energy - np.polyval(fit, separations))) print(f"energy vs. separation slope: {fit[0]:.4f} (should be > 0, roughly constant -- confinement)") print(f"deviation from a straight-line fit: {linfit_error:.2e}") .. rst-class:: sphx-glr-script-out .. code-block:: none energy vs. separation slope: 0.8862 (should be > 0, roughly constant -- confinement) deviation from a straight-line fit: 3.74e-07 .. GENERATED FROM PYTHON SOURCE LINES 90-99 The confinement signature itself: a straight line, not a Coulomb falloff ------------------------------------------------------------------------------ The animation shows the tube's *shape* stretching, but not the energetics that make it confinement rather than an ordinary field. Plotting the already-computed total field energy directly against separation makes the defining signature explicit: a straight line all the way out, in sharp contrast with an unconfined Coulomb-like field, whose energy would instead flatten out (approach a constant) at large separation as the field simply spreads into the extra room available. .. GENERATED FROM PYTHON SOURCE LINES 99-108 .. code-block:: Python fig2, ax2 = plt.subplots() ax2.plot(separations, total_energy, "o", label="total flux-tube energy") ax2.plot(separations, np.polyval(fit, separations), "k--", label=f"linear fit, slope={fit[0]:.3f}") ax2.set_xlabel("charge separation s") ax2.set_ylabel("total field energy") ax2.set_title("Confinement signature: energy grows linearly with separation") ax2.legend() fig2.tight_layout() .. image-sg:: /api/gallery/fields/gauge_confinement/images/sphx_glr_plot_flux_tube_confinement_002.png :alt: Confinement signature: energy grows linearly with separation :srcset: /api/gallery/fields/gauge_confinement/images/sphx_glr_plot_flux_tube_confinement_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.838 seconds) .. _sphx_glr_download_api_gallery_fields_gauge_confinement_plot_flux_tube_confinement.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_flux_tube_confinement.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_flux_tube_confinement.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_flux_tube_confinement.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_