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 Lopes† 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 † [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 | | 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 { } 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
R 3 0 0 0 0 4 2 Fi, with the total force being the ensemble aver- d r ˆr δ (r R)[u ˆr ] = u πR , valid for a gen- − − · 3 age of the cumulative force due to all the particles, eral three-dimensional vector u, we arrive at the D PN E i.e. Fb = Fi . Considering the gas to be simple expression, − i=1 in thermodynamic equilibrium, we arrive at the F r n r v ∇V , b( 0)= eq( 0) r0 (20) following expression: | which shows the local buoyant force to be pro- Z Z N 0 (3N)0 (3N) X ∂Up(ri,R) 0 portional to the local value of the external force Fb(r0)= d r d p ˆr (15) ∂r0 i field. This expression allows us to define a local i=1 i buoyancy force field (per unit volume), B(r0) = −βH r0+r ,{p },R e ( i 0 i ) F r /v { } . b( 0) and using Eq. (14) we arrive at the gen- × N!h3N Z(β, N; R) eralized Archimedes’ Principle, B(r)= ∇P (r), (21) Since the particles are non-interacting, Eq. (15) − can be further simplified to which is valid for any non-uniform pressure field. N Z 0 ∂ 0 (3) 0 −βV (r0+r ) 0 −βUp(r ,R) This shows that the pressure field is the effec- Fb(r0)= d r e ˆr 0 e . −βVeff ∂r tive scalar potential associated with the buoyancy (16) force. Before moving on to some specific exam- Finally, we may consider the rigid-probe limit ples of generalized buoyancy laws, we remark that (U0 ), such that the integral in the numer- V B(r0) is the buoyancy force that acts on any → ∞ ator of Eq. (16) becomes highly localized around immersed body of volume V , independently of the surface of the sphere. Then, we can approxi- its shape. However, this is only true provided ∂ 0 0 0 mate /∂r exp ( βUp (r ,R)) δ (r R) (see Ap- the later is small when compared to the scale of − ≈ − pendixA for further details) and perform a gradi- spatial-variations in the pressure field. For a larger ent expansion of exp ( βV (r)), i.e. immersed volume, its geometry starts influencing − 0 the balance of pressures and Archimedes’ principle −βV r0+r −βV (r0) 0 e ( ) e 1 βr ∇V . (17) ≈ − · |r0 strictly breaks down. This way, the average force reduces to the much VI. SOME ILLUSTRATIVE EXAMPLES simpler form, OF GENERALIZED BUOYANCY LAWS N −βV (r0) Fb(r0)= e (18) Previously, we have obtained a general expression −βVeff × Z for the buoyancy force caused in a small body im- (3) 0 0 0 0 mersed inside an ideal fluid under arbitrary exter- d r δ (r R) r βr ∇V r . − − · | 0 nal force fields. In this section, we will apply the previous expressions to some specific examples, to The physical interpretation of both terms in further elucidate the physical content of Eq. (21) Eq. (18) is very clear. The first term corresponds and provide non-standard examples of buoyancy. to a uniform pressure field and integrates to zero. As announced in the introduction, we will deter- As expected, in the absence of external force fields, mine the pressure and buoyancy force fields for the pressure is uniform across the system and does three paradigmatic cases: i) a fluid under a uni- not yield any buoyancy force. In contrast, the sec- form gravitational field, ii) a gas cloud confined by ond term is proportional to ∇V , which mea- |r0 a three-dimensional harmonic potential, and iii) a sures the effect of the lowest order inhomogeneity rotating fluid inside a cylinder under a centrifugal in the field at r0, i.e. a uniform pressure gradi- force. ent. For evaluating this term, the Dirac-δ can be 0 0 directly employed by recognizing that r = Rˆr , A. Uniform Gravitational Field giving a net force Z (3) 0 0 0 0 The simplest nontrivial case is that of a fluid on Fb(r0)=neq(r0) d r ˆr δ (r R) R ˆr ∇V − · |r0 a uniform gravitational field, i.e V (r) = V (z) = (19) mgz. To simplify the analysis, we will confine on the probe, where n(r0) is the equi- the gas to lie inside a column with a base area librium density of particles (as defined in A and an infinite height. By symmetry, the AppendixB). Finally, by using the identity equilibrium density of particles must only depend 6
U(r, R) n/nmax confining potential in its place, e.g. a three- 2 2 dimensional harmonic potential V (r) = 1/2mω r . Note that, despite allowing the gas particles to be located anywhere in space, this potential nat- urally defines an effective volume given by Veff = 2 3/2 2π/mω β . The equilibrium particle density (b) (a) is then shown to have a gaussian radial profile, r R r ξ 3 2 2 / / neq(r)=(N/Veff) exp ( /2)r /ξ , with a charac- p − 2 Figure 3. (a) Soft-wall potential of the probe for teristic width ξ = 3/mω β. Even though there increasing values of U0, approaching the rigid-probe are no walls, a pressure field can still be defined limit. The dashed lines represent the linear approxi- using Eq. 14, yielding mation to the potential near the spherical surface, i.e 2 r =R. (b) Equilibrium particle density profiles for the 3r P (r) = P0 exp (23) three examples considered. The curves are normal- −2ξ2 ized by their respective maxima and the distance is normalized to the characteristic lengths (defined, for where P0 = N/βVeff is the pressure at the origin each case, in the main text). and also the maximum value it can attain. Simi- larly, the buoyancy force field is obtained on the height (z) and is easily shown to decay 2 3P0 3r exponentially with characteristic distance z0 = B (r) = r exp ˆr, (24) −1 ξ2 −2ξ2 (βmg) which is also the average particle height. More precisely, the equilibrium particle density is being, unsurprisingly, directed in the outward ra- neq(z)=(N/Az0) exp( z/z0) and the pressure fol- − −z/z0 dial direction. lows the barometric law P (z) = P0e , where P0 =N/Az0β is the pressure at ground level z =0. Finally, the buoyancy force field can be easily ob- C. Centrifugal Buoyancy tained from Eq. (21), yielding As a last example, we consider the nontrivial P0 −z/z B (z)= e 0zˆ. (22) case of buoyancy inside a fluid confined to a ro- z0 tating cylindrical container of radius ξ and length This example can be made even more concrete by L. This system is assumed to be rotating around calculating the buoyancy force which is exerted in its revolution axis, with a fixed angular velocity a human being on the surface of the Earth. For Ω. This motion induces a centrifugal force that 2 2 is described by a potential V (ρ) = 1/2mΩ ρ , that purpose, we consider then average mass of − an air molecule (4.8 10−26kg), a temperature of where ρ is the radial cylindrical coordinate. This × 5 300K and a pressure at sea level of 10 N/m² (1 inverse harmonic potential is obviously not a con- atm), which estimates a typical variation length fining one, but this is not a problem as the later for the pressure field to be z0 8772 m. Since z0 is is provided by the cylinder’s hard wall. In this ≈ much larger than the height of a person, the gen- case, one can then define an effective volume, 2 1 2 2 Veff = (2πL)/(βmΩ ) exp /2βmΩ ξ 1 , such eralized Arquimedes’ principle applies and, given − the average volume of a person (6.2 10−2m3), one that the equilibrium particle density is neq(ρ) = × 1 2 2 concludes that a person would feel a total buoy- (N/Veff) exp /2βmΩ ρ inside the cylinder, and ancy force of 0.7 N. This is obviously a minor effect 0 outside (see Fig.3 b). Correspondingly, the that accounts for a reduction of about 0.1% in the buoyancy force field can be easily determined as perceived weight. follows B. Three-Dimensional Harmonic N 2 1 2 2 B (ρ) = βmΩ ρ exp β mΩ ρ ρˆ, (25) Confinement Potential −Veff 2
In truth, the formulas derived before do not re- which is now directed inwards. As in the previous quire any container for the pressure field to be situations, the natural tendency of a buoyant force well-defined. In fact, we can use any generic is to balance existing the centrifugal force acting upon any object floating inside. 7
VII. SUMMARY AND CONCLUDING edition of the international Physics competition REMARKS Plancks [1]. The participants’ responses included Many real-life physical situations, ranging from at- very physically rich and intuitive discussions. mospheric physics to buoyancy forces, can only be ACKNOWLEDGMENTS described by means of space-dependent pressure fields. From a microscopic point-of-view, it is not trivial to approach these thermodynamic concepts The authors acknowledge financial support without introducing relatively complex formula- by the Portuguese Foundation for Science and tions that obstruct an efficient teaching. In this Technology (FCT) within the Strategic Funding paper, we have attempted to bypass this problem UIDB/04650/2020 and COMPETE 2020 program and provide a renewed simple view on the micro- in FEDER component (European Union), through scopic definition of a pressure field inside an ideal Project No. POCI-01-0145-FEDER-028887. fluid that is acted upon by an arbitrary external S.M.J. and J.P.S.P. were further supported by force field. FCT Ph.D. grants PD/BD/142798/2018 and For this purpose, we relied on the standard PD/BD/142774/2018, respectively. The authors canonical formulation of statistical mechanics, and further thank Prof. J.M.B Lopes dos Santos, J. studied a gas model with independent classical Gonçalo Roboredo and J. Matos for fruitful dis- particles moving on an external scalar potential cussions. V(r). In order to unambiguously define the local pressure field, an additional short-ranged spheri- Appendix A: Useful Identities With Dirac-δ cal potential was considered, so as to provide the Functions needed point-like barometer capable of probing pressure inside the fluid. Using this setup, we In this section, we delve into more detail about have derived a closed-form formula for the pres- the trick used for the derivation of the local pres- sure field within the fluid, in the presence of ar- sure and the buoyant force field in SectIV andV. bitrary external potentials, assuming the limit of Specifically, we want to show that very small probes (compared to the variation scale of the external force field) with internal potentials ∂e−βUp ∂e−βUp that approach impenetrable spherical walls. With = (A1) this result, we arrived at a local version of the ideal ∂r − ∂R gas law, relating the value of the pressure field in a given point, to the equilibrium density of particles under reasonable assumptions. Recalling Eq. (1), in that same point. we begin by noting that Finally, we retraced the same process to calcu- ∂Up (r, R) −βU (r,R) 1 ∂ −βU (r,R) e p = e p . (A2) late the net force acting upon the local barome- ∂r −β ∂r ter and, with it, derived a generalized version of Archimedes’ principle capable of describing buoy- This function is highly localized around the sur- ancy in non-homogeneous force fields. We applied face of the sphere r = R. Let’s analyze how this this formulation to three paradigmatic cases: i) object acts inside the integrals we typically find: a fluid under a uniform gravitational field, ii) a Z ∞ ∂ −βUp(r,R) gas confined by a three-dimensional harmonic po- drg (r) e (A3) 0 ∂r tential, and iii) a rotating two-dimensional fluid under a centrifugal force. By its intrinsic rich- where g (r) is assumed to decay exponentially to ness but remarkable simplicity, we believe the for- zero as r increases. Such is the case when g (r) = mulation presented here to be a relevant resource e−βV (r). Using integration by parts, for teaching undergraduate students the connec- Z ∞ ∂ tions between statistical and thermodynamic de- dr e−βUp(r,R) g(r)= (A4) scriptions of fluids, allowing to consider situations 0 ∂r that go beyond traditional textbook material, but Z ∞ ∂ h i Z ∞ ∂ dr g (r) e−βUp(r,R) dre−βUp(r,R) g (r) , rather approach more realistic scenarios. Finally, 0 ∂r − 0 ∂r to assess the reaction of university level students, this microscopic formulation of pressure fields was the first term evaluates to zero because g ( ) = 0 −βU (r,R) ∞ used as a basis for a problem posed in the 2021 by assumption and e p = 0 when r < R in 8 the hard wall limit U0 . In this same limit, that appear in all the expressions. To be precise, → ∞ the second term simplifies to this quantity is found by evaluating the statistical Z ∞ ∂ Z ∞ ∂ average of the following characteristic function −βUp(r,R) dre g(r)= dr g (r)= g(R). N 0 ∂r R ∂r − X n(r)= δ (r ri) (B1) (A5) − Thus far, we have shown that i=1 Z ∞ ∂ drg (r) e−βU(r,R)=g (R) (A6) Considering only the gas subject to an external 0 ∂r potential (with no barometer present), its average in the canonical ensemble is for an arbitrary well-behaved function g (r). Then, 2 ! ∂ P pj /∂r exp [ βU (r, R)] = δ (r R) (3N) (3N) −β j +V(rj ) we can say that p R 0R P 2m − − d r d p iδ (r−ri) e under these conditions. Using the fact that r = R neq(r)= p2 (3N) (3N) P j under the action of δ (r R), we get that R 0R −β ( +V (rj )) − d r d pe j 2m ∂e−βUp R R ∂e−βUp (B2) =δ (r R)=δ (r R) = . ∂r − − r r ∂r which reduces to (A7) 0 N Finally, using the fact that R(∂Up/∂r ) = neq(r)= exp ( βV (r)) (B3) 0 Veff − r (∂Up/∂R), the original assertion, − ∂e−βUp ∂e−βUp and so the particles are more likely to be found in = (A8) ∂r − ∂R regions where the external potential V is smaller. In equilibrium, this is expected intuitively. If V (r) immediately follows. is constant, then the density of particles reduces N/V Appendix B: Equilibrium Density of particles to the familiar expression . in the Presence of an External Field
In this appendix, we derive the equilibrium den- sity of particles for an ideal classical gas under the influence of an arbitrary scalar potential V(r). When deriving the expressions of the pressure and buoyancy force fields, this is a necessary quantity
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