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A New Microscopic Look Into Pressure Fields and Local Buoyancy

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A New Microscopic Look Into Pressure Fields and Local Buoyancy A New Microscopic Look into Pressure Fields and Local Buoyancy S. M. João,∗ J. P. Santos Pires, and J. M. Viana Parente Lopesy Centro de Física das Universidades do Minho e Porto and Faculty of Sciences, University of Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal The concept of local pressure is pivotal to describe many important physical phenomena, such as buoyancy or atmospheric phenomena, which always require the consideration of space-varying pres- sure fields. These fields have precise definitions within the phenomenology of hydro-thermodynamics, but a simple and pedagogical microscopic description based on Statistical Mechanical is still lacking in present literature. In this paper, we propose a new microscopic definition of the local pressure field inside a classical fluid, relying on a local barometer potential that is built into the many-particle Hamiltonian. Such a setup allows the pressure to be locally defined, at an arbitrary point inside the fluid, simply by doing a standard ensemble average of the radial force exerted by the barom- eter potential on the gas’ particles. This setup is further used to give a microscopic derivation of the generalized Archimedes’s buoyancy principle, in the presence of an arbitrary external field. As instructive examples, buoyancy force fields are calculated for ideal fluids in the presence of: i) a uniform force field, ii) a spherically symmetric harmonic confinement field, and iii) a centrifugal rotating frame. I. INTRODUCTION amples [5, 17]. To address this problem, it be- comes necessary for the community of Physics Thermodynamics is an essential tool for un- teachers to implement new ways of explaining derstanding the world around us, giving a self- these concepts and, in particular, devise a wider contained but simple description of physical pro- variety of examples that more clearly connect cesses involving many particles, but using a min- to real-life situations, whilst retaining a simple imal number of variables. For example, temper- enough description that makes their physical in- ature and pressure are pivotal concepts in the terpretation as transparent as possible. thermodynamics of fluids but are also intrinsically In standard textbook treatments, pressure is de- macroscopic, emerging only if the number of par- fined as the average force (per unit area) caused ticles is very large and maintained in some state by microscopic particles that collide elastically of equilibrium. with the walls of the fluid’s container. In this A connection between the basic laws that gov- sense, it is a global property of any fluid in ther- ern the mechanics of microscopic degrees of free- modynamic equilibrium, when confined to a fi- dom and their thermodynamic manifestation is nite volume. In contrast, many physical phe- provided by the edifice of Statistical Physics [8, nomena are known to depend on a local defini- 19, 20]. In its simplest description, the temper- tion of pressure, described as a position-dependent ature of a fluid is seen as a measure of the mean scalar field that exists within the system. At- energy per particle, while pressure describes the mospheric physics [2] and hydrodynamics [12] are average momentum transferred from the jiggling two important and paradigmatic examples where particles to the walls of its container by means non-uniform pressure fields are ubiquitous con- of random elastic collisions. Despite providing a cepts. Nonetheless, a simpler example is pro- seemingly clear-cut explanation of the microscopic vided by the well-known phenomenon of buoyancy foundations of thermodynamics, several investiga- (Archimedes’ Principle), which relies upon the ex- tions on physics education [3,5,9, 10, 15–17] have istence of unbalanced pressures acting on different arXiv:2106.10369v1 [physics.class-ph] 18 Jun 2021 proved these concepts difficult to grasp for most surfaces of an immersed solid body. This position- physics/engineering students, especially when re- dependent pressure is caused by the Earth’s grav- quired to provide insight on practical situations itational field, but generally it may be due to any that fall outside the scope of typical textbook ex- other external field that the fluid particles expe- rience. In the literature, a myriad of nonstan- dard derivations [4,6,7, 11, 13, 22] and general- izations [18, 21, 23] of Arquimedes’ principle can ∗ [email protected] be found, some of which going beyond the case y [email protected] of a non-interacting classical gas. Nonetheless, we 2 describe the confinement of the gas particles inside F a container or reflect an external force field act- ing globally upon the system (e.g. a gravitational field). In order to have a consistent definition of a position-dependent pressure field inside the sys- U(r) z′ tem, one is forced to introduce some kind of local y′ v probe. This is analogous to the more common use z r0 x′ of test electrical charges to define electromagnetic y fields in space. Therefore, we add to the Hamil- x R tonian of the gas a short-ranged perturbation po- tential Up(r) representing a small spherical probe of radius R (as depicted in Fig.1). The probe is V(r) assumed to be centered in an arbitrary but fixed position r = r0 (where the pressure is measured), 0 and r = r r0 is taken to be a position vector − Figure 1. Sketch of the physical situation. referenced to the center of the probe. In the present context, we define the pressure at have not found pedagogical presentations in the position r0 as the average momentum transferred literature that illustrate how scalar pressure fields from the gas particles to the spherical probe, by can emerge from a microscopic theory. The goal of means of collisions. To estimate this quantity, it is this work is to provide such a description based on useful to model the probe’s surface, not as a rigid a kinetic theory of classical non-interacting par- spherical wall, but rather as a soft-wall potential ticles, common knowledge for an undergraduate which allows the particles to slightly penetrate be- student. fore being repelled outwards. This means that any In this paper, we develop a simple microscopic gas particle, at a distance r0 = r0 from the probe’s j j description of non-uniform pressure fields in ideal center will feel a spherically symmetric potential, fluids, based solely on a standard canonical for- 0 0 0 r R r R mulation of Statistical Physics, as well as rather Up(r ;R)=U0f − ΘH − (1) R − R simple kinetic theory arguments. We define the local pressure by introducing a fictitious localized where U0 measures the strength of the interaction, spherical “soft-wall” potential that serves as a test- ΘH(x) is the Heaviside step function and f(x) is a ing probe. This probe plays the role of a small continuous and monotonous function that defines barometer, whose volume couples to the local pres- the overall shape of the soft-wall potential, and sure, defining a pressure field. Assuming a ther- behaves as f(x 1−) x close to the surface of ≈ ≈− malized gas, we derive a closed-form expression the sphere (see Fig.3a). Such a potential implies for the pressure field in the presence of an arbi- that the gas particles feel a roughly uniform out- trary external potential and, with it, describe the wards radial force inside the sphere, but no force buoyant force acting upon a small body immersed outside of it. With the above definitions, the full in this fluid. Finally, this generalized Archimedes’ Hamiltonian of the gas reads principle is made concrete by the application to N 2 X pi three simple examples: i) a fluid under a uniform ( ri; pi ;R)= +V (ri)+Up(ri r0;R) ; H f g 2m − gravitational field, ii) a gas cloud confined by a i=1 three-dimensional harmonic potential, and iii) a (2) rotating two-dimensional fluid under a centrifugal where (ri; pi) are the phase-space coordinates of force. the corresponding particles. Note that, for sim- plicity, we have considered the gas to consist of II. MICROSCOPIC MODEL OF AN IDEAL GAS monoatomic particles that have no (active) inter- nal degrees of freedom. Aiming at a microscopic description of a fluid III. CANONICAL ENSEMBLE pressure field, we start with a system having FORMULATION a large number N of independent and non- relativistic classical particles of mass m, that move Having established our working microscopic across a scalar potential V (r). This potential can model, we are now in position to perform a stan- 3 dard Statistical Mechanical analysis of this non- interacting gas. Concretely, we will work in ∆<latexit sha1_base64="xtD8q2cqGRdlXnqtqtirz3mkwWc=">AAAB+3icbVDLSsNAFJ3UV62vWpduBovgqiSi6LI+Fi4r2Ac0oUymk3boZBJmbsQS8ituXCji1h9x5984abPQ1gMDh3Pu5Z45fiy4Btv+tkorq2vrG+XNytb2zu5edb/W0VGiKGvTSESq5xPNBJesDRwE68WKkdAXrOtPbnK/+8iU5pF8gGnMvJCMJA84JWCkQbXm3jIBBLshgTElIr3KBtW63bBnwMvEKUgdFWgNql/uMKJJyCRQQbTuO3YMXkoUcCpYVnETzWJCJ2TE+oZKEjLtpbPsGT42yhAHkTJPAp6pvzdSEmo9DX0zmUfUi14u/uf1EwguvZTLOAEm6fxQkAgMEc6LwEOuGAUxNYRQxU1WTMdEEQqmroopwVn88jLpnDac84Z9f1ZvXhd1lNEhOkInyEEXqInuUAu1EUVP6Bm9ojcrs16sd+tjPlqyip0D9AfW5w+xhpQ4</latexit> the canonical ensemble (at a finite temperature A <latexit sha1_base64="SZdh5QT1NTY7zEHN0WSihpxhBNI=">AAAB8nicbVBNS8NAFHypX7V+VT16WSyCp5KIoseqF48VrC2koWy223bpZhN2X4QS+jO8eFDEq7/Gm//GTZuDtg4sDDPvsfMmTKQw6LrfTmlldW19o7xZ2dre2d2r7h88mjjVjLdYLGPdCanhUijeQoGSdxLNaRRK3g7Ht7nffuLaiFg94CThQUSHSgwEo2glvxtRHDEqs+tpr1pz6+4MZJl4BalBgWav+tXtxyyNuEImqTG+5yYYZFSjYJJPK93U8ISyMR1y31JFI26CbBZ5Sk6s0ieDWNunkMzU3xsZjYyZRKGdzCOaRS8X//P8FAdXQSZUkiJXbP7RIJUEY5LfT/pCc4ZyYgllWtishI2opgxtSxVbgrd48jJ5PKt7F3X3/rzWuCnqKMMRHMMpeHAJDbiDJrSAQQzP8ApvDjovzrvzMR8tOcXOIfyB8/kDcaCRWw==</latexit>
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