stage, averaged relative error of the HOSVD-TROM finite element solutions, and
statistics of relative gain (79) for $\nu=0.01$, $K=20\times 2^{8}=5,120$.

$\widetilde{\varepsilon}=10^{-3}$

HOSVD ranks | [76,2,2,2,2,2,2,2,2,11,12] |  $\\#(\mbox{online}(\widetilde{\boldsymbol{\Phi}}))$=2568444
---|---|---
TT ranks | [75,77,79,76,77,75,71,67,61,11] |  $\\#(\mbox{online}(\widetilde{\boldsymbol{\Phi}}))$=100747
$n$ | 5 | 8 | 10
TROM FE error |  |  |
$L^{\infty}(0,T;L^{2}(\Omega))$ | 3.45e-2 | 7.07e-3 | 5.58e-3
$L^{2}(0,T;H^{1}(\Omega))$ | 5.59e-2 | 1.10e-2 | 5.86e-3
$G_{X}$ | mean | std | min | mean | std | min | mean | std | min
HOSVD | 6.95 | 1.10 | 5.41 | 22.56 | 6.97 | 12.02 |  32.66 |  24.70 | 10.00
TT | 6.95 | 1.09 | 5.41 | 22.54 | 6.94 | 11.97 |  31.83 |  23.58 | 9.99

$\widetilde{\varepsilon}=10^{-5}$

HOSVD ranks | [184,2,2,2,2,2,2,2,2,15,18] |  $\\#(\mbox{online}(\widetilde{\boldsymbol{\Phi}}))$=12718412
---|---|---
TT ranks | [183,235,288,305,319,295,246,187,128,18] |  $\\#(\mbox{online}(\widetilde{\boldsymbol{\Phi}}))$=1110964
$n$ | 5 | 10 | 15
TROM FE error |  |  |
$L^{\infty}(0,T;L^{2}(\Omega))$ | 3.45e-2 | 5.56e-3 | 5.51e-3
$L^{2}(0,T;H^{1}(\Omega))$ | 5.59e-2 | 5.80e-3 | 5.08e-3
$G_{X}$ | mean | std | min | mean | std | min | mean | std | min
HOSVD | 6.95 | 1.10 | 5.41 | 33.34 | 25.90 | 10.01 |  19.45 |  16.33 | 5.60
TT | 6.95 | 1.10 | 5.41 | 33.34 | 25.90 | 10.01 |  19.45 |  16.33 | 5.60

$\widetilde{\varepsilon}=10^{-7}$

HOSVD ranks | [476,2,2,2,2,2,2,2,2,18,25] |  $\\#(\mbox{online}(\widetilde{\boldsymbol{\Phi}}))$=54835592
---|---|---
TT ranks | [528,618,753,858,866,725,520,341,211,25] |  $\\#(\mbox{online}(\widetilde{\boldsymbol{\Phi}}))$=6975287
$n$ | 5 | 10 | 15
TROM FE error |  |  |
$L^{\infty}(0,T;L^{2}(\Omega))$ | 3.44e-2 | 5.56e-3 | 5.51e-3
$L^{2}(0,T;H^{1}(\Omega))$ | 5.59e-2 | 5.80e-3 | 5.08e-3
$G_{X}$ | mean | std | min | mean | std | min | mean | std | min
HOSVD | 6.95 | 1.10 | 5.41 | 33.34 | 25.90 | 10.01 |  19.45 |  16.33 | 5.60
TT | 6.95 | 1.10 | 5.41 | 33.34 | 25.90 | 10.01 |  19.45 |  16.33 | 5.60

Table 8 shows tensor compression ranks, number of elements passed to the
online stage, the averaged relative error of the HOSVD-TROM finite element
solutions, the relative gain statistics for three different levels of tensor
compression error $\widetilde{\varepsilon}=10^{-3},10^{-5},10^{-7}$ with three
different reduced space dimensions $n$ for each case. We observe that it is
possible to achieve performance that is very close to the best one with
$\widetilde{\varepsilon}$ as large as $10^{-3}$, provided $n$ is large enough.
Similarly to the results for $\nu=0.1$, we suggest that for the given problem,
discretization and parameter sampling, the finite element error is dominated
by the interpolation error of the TROM and using a tighter compression
threshold or larger $n$ does not lead to more accurate ROM solutions. It seems
beneficial to use low-accuracy tensor decompositions to save on both the
computation and storage, while not losing much in terms of relative gain
compared to more expensive options. As expected, the TT format becomes more
cost-efficient for the higher parameter space dimension $D$.

Overall, while the accuracy increase of HOSVD- and TT-TROM over POD-ROM is
still substantial in the advection-diffusion setting with $D=9$ parameters, it
is smaller than the one for the heat equation considered in Section 5.2. This
is most probably caused by larger variability of the snapshots with respect to
parameter variation making the problem a good candidate for non-interpolatory
TROM (not studied here).

##  6 Conclusions

Summarizing the findings of the paper, the tensorial projection ROM for
parametric dynamical systems builds on several new ideas:
(i) To approximately represent the set of observed snapshots, it uses low-rank
tensor formats, rather than a truncated SVD of the snapshot matrix. The
corresponding tensor decompositions provide POD-type universal basis while
preserving information about solution variation with respect to parameters.
(ii) This additional information is used to find a local (parameter-specific)
ROM basis for any incoming parameter that is not necessarily from the
training/sampling set.
(iii) The local basis can be represented by its coordinates in the universal
low-dimensional basis allowing an effective split of the ROM evaluation
between the online and offline phases.

An interpolation procedure was suggested to extract the information about
parameter dependence of the solutions, and thus of the ROM spaces, from the
low-rank tensor decompositions. Online stage uses fast linear algebra with
complexity depending only on the compression ranks. Non-interpolatory or
hybrid approaches are also possible and in fact can produce even more accurate
and robust TROMs. We will study these options elsewhere. For interpolatory
TROMs, Theorem 1 proves an estimate on the representation power of the local
ROM bases. Numerical experiment with parameterized heat equation supported the
estimate and illustrated the role of each of its terms.

Three popular compressed tensor formats were considered to represent the low-
rank tensor in the TROM. Of course, other low-rank tensor decompositions can
be used within the general framework of TROM. Out of the three tested, we
found HOSVD to be most user-friendly and cost-efficient provided either the
dimension of the parameter space is not too large or no Cartesian structure is
exploited in organizing the snapshots. Otherwise, TT-TROM provides necessary
tools to handle higher-dimensional parameter spaces. We also observed that the
accuracy of TROMs crucially depend on $n$, $\widetilde{\varepsilon}$ and
parameter domain sampling, but not as much on the particular low-rank tensor
format employed.

Finally, for higher-dimensional parameter spaces a grid-based sampling of the
parameter domain becomes prohibitively expensive in terms of offline
computation costs. Significant offline costs also incur for problems with less
smooth dependence of solution on parameters, which would require a denser
sampling, and for problems where each high-fidelity solve is expensive because
of fine spatial or temporal resolution. We see several ways to develop TROMs
addressing these challenges: (i) use a sophisticated sampling, e.g., based on
a greedy strategy, and organize the snapshots in 3D tensors, (ii) to benefit
from Cartesian structure and higher order tensor decompositions, apply a
tensor completion method to find a low-rank representation of the snapshot
tensor sampled at a few nodes of the parameter grid, and (iii) combine TROMs
with compressed formats to represent high-fidelity snapshots. We leave these
options for a future research.

## Acknowledgments

M.O. was supported in part by the U.S. National Science Foundation under
awards DMS-2011444 and DMS-1953535. This material is based upon research
supported in part by the U.S. Office of Naval Research under award number
N00014-21-1-2370 to A.M. The authors thank Vladimir Druskin, Traian Iliescu,
and Vladimir Kazeev for their comments on the first draft of this paper.

## References

  * [1] D. Amsallem and C. Farhat, Interpolation method for adapting reduced-order models and application to aeroelasticity, AIAA journal, 46 (2008), pp. 1803–1813.
  * [2] D. Amsallem, M. J. Zahr, and C. Farhat, Nonlinear model order reduction based on local reduced-order bases, International Journal for Numerical Methods in Engineering, 92 (2012), pp. 891–916.
  * [3] A. C. Antoulas, D. C. Sorensen, and S. Gugercin, A survey of model reduction methods for large-scale systems, tech. report, 2000.
  * [4] J. Ballani, D. Kressner, and M. D. Peters, Multilevel tensor approximation of pdes with random data, Stochastics and Partial Differential Equations: Analysis and Computations, 5 (2017), pp. 400–427.
  * [5] U. Baur, C. Beattie, P. Benner, and S. Gugercin, Interpolatory projection methods for parameterized model reduction, SIAM Journal on Scientific Computing, 33 (2011), pp. 2489–2518.
  * [6] J. A. Bengua, H. N. Phien, H. D. Tuan, and M. N. Do, Efficient tensor completion for color image and video recovery: Low-rank tensor train, IEEE Transactions on Image Processing, 26 (2017), pp. 2466–2479.
  * [7] P. Benner, S. Dolgov, A. Onwunta, and M. Stoll, Low-rank solvers for unsteady stokes–brinkman optimal control problem with random data, Computer Methods in Applied Mechanics and Engineering, 304 (2016), pp. 26–54.
  * [8] , Solving optimal control problems governed by random navier-stokes equations using low-rank methods, arXiv preprint arXiv:1703.06097, (2017).
  * [9] P. Benner and L. Feng, A robust algorithm for parametric model order reduction based on implicit moment matching, in Reduced order methods for modeling and computational reduction, Springer, 2014, pp. 159–185.
  * [10] P. Benner, S. Gugercin, and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM review, 57 (2015), pp. 483–531.
  * [11] P. Benner, A. Onwunta, and M. Stoll, Low-rank solution of unsteady diffusion equations with stochastic coefficients, SIAM/ASA Journal on Uncertainty Quantification, 3 (2015), pp. 622–649.
  * [12] S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, second ed., 2002.
  * [13] S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proceedings of the national academy of sciences, 113 (2016), pp. 3932–3937.
  * [14] T. Bui-Thanh, K. Willcox, and O. Ghattas, Model reduction for large-scale systems with high-dimensional parametric input space, SIAM Journal on Scientific Computing, 30 (2008), pp. 3270–3288.
  * [15] K. Carlberg, C. Farhat, J. Cortial, and D. Amsallem, The gnat method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows, Journal of Computational Physics, 242 (2013), pp. 623–647.
  * [16] J. D. Carroll and J.-J. Chang, Analysis of individual differences in multidimensional scaling via an n-way generalization of “eckart-young” decomposition, Psychometrika, 35 (1970), pp. 283–319.
  * [17] S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM Journal on Scientific Computing, 32 (2010), pp. 2737–2764.
  * [18] F. Chinesta, A. Ammar, and E. Cueto, Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models, Archives of Computational methods in Engineering, 17 (2010), pp. 327–350.
  * [19] F. Chinesta, R. Keunings, and A. Leygue, The proper generalized decomposition for advanced numerical simulations: a primer, Springer Science & Business Media, 2013.
  * [20] A. Cohen, R. Devore, and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic pde’s, Analysis and Applications, 9 (2011), pp. 11–47.
  * [21] L. De Lathauwer, B. De Moor, and J. Vandewalle, A multilinear singular value decomposition, SIAM journal on Matrix Analysis and Applications, 21 (2000), pp. 1253–1278.
  * [22] V. De Silva and L.-H. Lim, Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM Journal on Matrix Analysis and Applications, 30 (2008), pp. 1084–1127.
  * [23] S. V. Dolgov, V. A. Kazeev, and B. N. Khoromskij, Direct tensor-product solution of one-dimensional elliptic equations with parameter-dependent coefficients, Mathematics and computers in simulation, 145 (2018), pp. 136–155.
  * [24] M. Drohmann, B. Haasdonk, and M. Ohlberger, Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation, SIAM Journal on Scientific Computing, 34 (2012), pp. A937–A969.
  * [25] J. L. Eftang, D. J. Knezevic, and A. T. Patera, An hp certified reduced basis method for parametrized parabolic partial differential equations, Mathematical and Computer Modelling of Dynamical Systems, 17 (2011), pp. 395–422.
  * [26] J. L. Eftang, A. T. Patera, and E. M. Rønquist, An” hp” certified reduced basis method for parametrized elliptic partial differential equations, SIAM Journal on Scientific Computing, 32 (2010), pp. 3170–3200.
  * [27] M. Eigel, M. Pfeffer, and R. Schneider, Adaptive stochastic galerkin fem with hierarchical tensor representations, Numerische Mathematik, 136 (2017), pp. 765–803.
  * [28] S. Gandy, B. Recht, and I. Yamada, Tensor completion and low-n-rank tensor recovery via convex optimization, Inverse problems, 27 (2011), p. 025010.
  * [29] L. Grasedyck, D. Kressner, and C. Tobler, A literature survey of low-rank tensor approximation techniques, GAMM-Mitteilungen, 36 (2013), pp. 53–78.
  * [30] M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, ESAIM: Mathematical Modelling and Numerical Analysis, 41 (2007), pp. 575–605.
  * [31] M. A. Grepl and A. T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, ESAIM: Mathematical Modelling and Numerical Analysis, 39 (2005), pp. 157–181.
  * [32] M. Griebel and H. Harbrecht, Analysis of tensor approximation schemes for continuous functions, Foundations of Computational Mathematics, (2021), pp. 1–22.
  * [33] S. Gugercin and A. C. Antoulas, A survey of model reduction by balanced truncation and some new results, International Journal of Control, 77 (2004), pp. 748–766.
  * [34] W. Hackbusch, Tensor spaces and numerical tensor calculus, vol. 42, Springer, 2012.
  * [35] W. Hackbusch and B. N. Khoromskij, Tensor-product approximation to operators and functions in high dimensions, Journal of Complexity, 23 (2007), pp. 697–714.
  * [36] R. A. Harshman, Foundations of the parafac procedure: Models and conditions for an” explanatory” multimodal factor analysis, (1970).
  * [37] P. Hartman, Ordinary Differential Equations, vol. 590, John Wiley and Sons, 1964.
  * [38] J. Håstad, Tensor rank is np-complete, Journal of Algorithms, 11 (1990), pp. 644–654.
  * [39] J. S. Hesthaven, G. Rozza, and B. Stamm, Certified reduced basis methods for parametrized partial differential equations, vol. 590, Springer, 2016\.
  * [40] F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of products, Journal of Mathematics and Physics, 6 (1927), pp. 164–189.
  * [41] B. Huang, C. Mu, D. Goldfarb, and J. Wright, Provable low-rank tensor recovery, Optimization-Online, 4252 (2014), pp. 455–500.
  * [42] S. Kastian, D. Moser, L. Grasedyck, and S. Reese, A two-stage surrogate model for neo-hookean problems based on adaptive proper orthogonal decomposition and hierarchical tensor approximation, Computer Methods in Applied Mechanics and Engineering, 372 (2020), p. 113368.
  * [43] G. Kerschen, J.-c. Golinval, A. F. Vakakis, and L. A. Bergman, The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview, Nonlinear dynamics, 41 (2005), pp. 147–169.
  * [44] B. N. Khoromskij and C. Schwab, Tensor-structured galerkin approximation of parametric and stochastic elliptic pdes, SIAM Journal on Scientific Computing, 33 (2011), pp. 364–385.
  * [45] H. A. Kiers, Towards a standardized notation and terminology in multiway analysis, Journal of Chemometrics: A Journal of the Chemometrics Society, 14 (2000), pp. 105–122.
  * [46] T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM review, 51 (2009), pp. 455–500.
  * [47] B. Kramer and K. E. Willcox, Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition, AIAA Journal, 57 (2019), pp. 2297–2307.
  * [48] D. Kressner and C. Tobler, Low-rank tensor krylov subspace methods for parametrized linear systems, SIAM Journal on Matrix Analysis and Applications, 32 (2011), pp. 1288–1316.
  * [49] K. Lee, H. C. Elman, and B. Sousedik, A low-rank solver for the navier–stokes equations with uncertain viscosity, SIAM/ASA Journal on Uncertainty Quantification, 7 (2019), pp. 1275–1300.
  * [50] Y. Liang, H. Lee, S. Lim, W. Lin, K. Lee, and C. Wu, Proper orthogonal decomposition and its applications—part i: Theory, Journal of Sound and vibration, 252 (2002), pp. 527–544.
  * [51] , Proper orthogonal decomposition and its applications—part i: Theory, Journal of Sound and vibration, 252 (2002), pp. 527–544.
  * [52] Y. Liang, W. Lin, H. Lee, S. Lim, K. Lee, and H. Sun, Proper orthogonal decomposition and its applications–part ii: Model reduction for mems dynamical analysis, Journal of Sound and Vibration, 256 (2002), pp. 515–532.
  * [53] J. Liu, P. Musialski, P. Wonka, and J. Ye, Tensor completion for estimating missing values in visual data, IEEE transactions on pattern analysis and machine intelligence, 35 (2012), pp. 208–220.
  * [54] J. L. Lumley, The structure of inhomogeneous turbulent flows, Atmospheric turbulence and radio wave propagation, (1967).
  * [55] A. Nouy, Low-rank tensor methods for model order reduction, arXiv preprint arXiv:1511.01555, (2015).
  * [56] , Low-rank methods for high-dimensional approximation and model order reduction, Model reduction and approximation, P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, eds., SIAM, Philadelphia, PA, (2017), pp. 171–226.
  * [57] I. Oseledets and E. Tyrtyshnikov, Tt-cross approximation for multidimensional arrays, Linear Algebra and its Applications, 432 (2010), pp. 70–88.
  * [58] I. V. Oseledets, Tensor-train decomposition, SIAM Journal on Scientific Computing, 33 (2011), pp. 2295–2317.
  * [59] A. T. Patera, G. Rozza, et al., Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations, 2007\.
  * [60] M. Rathinam and L. R. Petzold, A new look at proper orthogonal decomposition, SIAM Journal on Numerical Analysis, 41 (2003), pp. 1893–1925.
  * [61] C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition, International Journal of Bifurcation and Chaos, 15 (2005), pp. 997–1013.
  * [62] R. Schneider and A. Uschmajew, Approximation rates for the hierarchical tensor format in periodic sobolev spaces, Journal of Complexity, 30 (2014), pp. 56–71.
  * [63] N. D. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. E. Papalexakis, and C. Faloutsos, Tensor decomposition for signal processing and machine learning, IEEE Transactions on Signal Processing, 65 (2017), pp. 3551–3582.
  * [64] L. Sirovich, Turbulence and the dynamics of coherent structures. i. coherent structures, Quarterly of applied mathematics, 45 (1987), pp. 561–571.
  * [65] N. T. Son, A real time procedure for affinely dependent parametric model order reduction using interpolation on grassmann manifolds, International Journal for Numerical Methods in Engineering, 93 (2013), pp. 818–833.
  * [66] V. N. Temlyakov, Estimates for the best bilinear approximations of periodic functions, Trudy Matematicheskogo Instituta imeni VA Steklova, 181 (1988), pp. 250–267.
  * [67] L. Trefethen, Multivariate polynomial approximation in the hypercube, Proceedings of the American Mathematical Society, 145 (2017), pp. 4837–4844.
  * [68] L. R. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, 31 (1966), pp. 279–311.
  * [69] M. Yuan and C.-H. Zhang, On tensor completion via nuclear norm minimization, Foundations of Computational Mathematics, 16 (2016), pp. 1031–1068.


Projects/QSM22/Paper3/fountain.tex

# On the Fountain Effect in Superfluid Helium

Phil Attard<EMAIL_ADDRESS>
###### Abstract

The origin of the fountain pressure that arises from a small temperature
difference in superfluid helium is analyzed. The osmotic pressure explanation
due to Tisza (1938), based on the different fractions of ground state bosons,
is formulated as an ideal solution and shown to overestimate the fountain
pressure by an order of magnitude or more. The experimentally confirmed
thermodynamic expression of H. London (1939), in which the temperature
derivative of the pressure equals the entropy per unit volume, is shown to be
equivalent to equality of chemical potential, not chemical potential divided
by temperature. The former results from minimizing the energy at constant
entropy, whereas the latter results from maximizing the entropy. It is
concluded that superfluid flow in the fountain effect is driven by mechanical,
not statistical, forces. This new principle appears to be a general law for
flow in superfluids and superconductors.

###### pacs:

##  I Introduction

One of the earliest and perhaps still the most spectacular manifestation of
superfluidity is the fountain effect. In this helium vigorously spurts from
the open end of a heated tube that is connected by a capillary or microporous
frit to a chamber of liquid helium maintained below the condensation
temperature.Allen38 ; Balibar17  Closing the heated chamber and measuring the
fountain pressure is a common experimental technique for obtaining the
entropy.Donnelly98

The commonly accepted physical explanation for the fountain effect as an
osmotic pressure is due to Tisza.Balibar17 ; Tisza38  It is based on his two-
fluid model, which in turn is based on F. London’s proposal that the
$\lambda$-transition in liquid helium is due to Bose-Einstein
condensation.FLondon38  The two-fluid model says that helium consists of a
mixture of He I and He II, which are excited state and ground state bosons,
respectively. The latter comprise the superfluid and their fraction increases
with decreasing temperature below the condensation temperature. This gives
rise to the notion that osmosis drives He II selectively through the capillary
from the low temperature, high concentration chamber to the high temperature,
low concentration one. In general in a binary solution the mixing entropy
favors concentration equality, and this is what is said to create the osmotic
pressure.

H. LondonHLondon39  carried out a thermodynamic analysis of the fountain
effect that made a quantitative prediction for the pressure difference for a
given temperature difference. His result is historically important as the
quantitative experimental verification of this formula was evidence for the
picture of superfluid helium being in a state of zero entropy. To the present
day his expression remains significant as fountain pressure measurements are
used to establish benchmark results against which calorimetric methods for the
entropy of helium may be tested.Donnelly98  The results obtained from his
expression are also used as calibration standards for instruments.Hammel61

Although this paper does not go beyond the expression of H. London,HLondon39
it does shed significant new light on the physical origin of the fountain
effect and on the nature of superfluidity. The major result is to show that
the H. London expression is obtained by minimizing the energy of the system
with respect to particle transfer between the two chambers, Sec. II.1.
Conversely, it shows that the result is inconsistent with maximizing the
entropy of the system. This apparent violation of the Second Law of
Thermodynamics is peculiar to superfluid flow.

The new derivation and interpretation of the H. London expression given here
has a number of consequences for superfluid flow. One is that it rules out the
osmotic pressure explanation of the fountain effect due to Tisza.Tisza38  The
latter is statistical and based on the mixing entropy, which contradicts the
present finding that it is the minimization of energy that underlies the
fountain effect. To underscore this point, in Sec. III quantitative
calculations are made of the osmotic prediction for the fountain pressure
using both calculatedAttard21 ; Attard22a  and measuredDonnelly98  data for
the condensed boson fraction. It is shown that the predicted fountain
pressures do not agree with measured values.Hammel61  Although these
calculations are made at the level of ideal solution theory, the quantitative
error is so large, and the qualitative difference with energy minimization so
stark, that the osmotic explanation for the fountain effect appears nonviable.

This paper is the fourth in a series on superfluidity and superconductivity.
Attard22a ; Attard22b ; Attard22c  The general formulation of quantum
statistical mechanics that underlies them is given in Ref. Attard21, . The
author has also given coherent formulations of equilibriumTDSM  and non-
equilibriumNETDSM  thermodynamics and classical statistical mechanics, which
results and concepts are freely used below.

##  II Thermodynamic Analysis

###  II.1 The Fountain Pressure Minimizes Energy

The aim of this section is show that the expression of H. LondonHLondon39  for
the fountain pressure follows from the minimization of the energy of the
subsystems.

Consider two closed chambers of helium, $A$ and $B$, each in contact with its
own thermal reservoir of temperature $T_{A}$ and $T_{B}$, and having pressure
$p_{A}$ and $p_{B}$. The chambers are connected by a capillary through which
superfluid, and only superfluid, flows. Chamber $A$ in practice is at the
lower temperature, and consists of saturated liquid and vapor, but these
points are presently unimportant. As H. London points out,HLondon39  in the
optimum steady state the pressure of the second chamber is a function of its
temperature and the pressure and temperature of the first chamber,
$p_{B}=p(T_{B};p_{A},T_{A})$.

The result given by H. LondonHLondon39  says that the derivative of the
pressure of the second chamber with respect to its temperature for fixed first
chamber equals the entropy density,HLondon39

$\frac{\mathrm{d}p_{B}}{\mathrm{d}T_{B}}=\rho_{B}s_{B}.$ (2.1)

Here $\rho$ is the number density and $s$ is the entropy per particle. (In
this paper I use lower case letters to denote quantities per particle; H.
LondonHLondon39  uses them to denote quantities per unit mass.) (The
derivative of the pressure and like quantities with respect to temperature for
fixed first chamber parameters are total derivatives, which means that their
integral along the fountain path can be evaluated analytically. This is useful
in the analysis of the H. London derivationHLondon39  in Appendix A.)

Whereas H. LondonHLondon39  purported to derive this result using a work-heat
flow cycle (see Appendix A), here I seek to locate it within the broader
principles of thermodynamics. To this end there is a subtle point regarding
extensivity that needs to be clarified.

In equilibrium thermodynamics it is generally recognized that quantities such
as energy, entropy, and free energy are extensive, which means that the total
value is the sum of the individual values for independent or quasi-independent
subsystems. For the present case of two subsystems of different temperatures
this general rule does not hold for the free energy. This is easily seen by
considering the two to be quasi-independent such that the probability of a
joint state is the product of the individual probabilitiesTDSM

$\displaystyle\wp(X_{A},X_{B})$ $\displaystyle=$
$\displaystyle\wp_{A}(X_{A})\wp_{B}(X_{B})$ (2.2) $\displaystyle\propto$
$\displaystyle e^{S_{A}(X_{A})/k_{\mathrm{B}}}e^{S_{B}(X_{B})/k_{\mathrm{B}}}$
$\displaystyle=$ $\displaystyle
e^{-F_{A}(X_{A})/k_{\mathrm{B}}T_{A}}e^{-F_{B}(X_{B})/k_{\mathrm{B}}T_{B}}.$

Here $k_{\mathrm{B}}$ is Boltzmann’s constant, and $S$ and $F$ are the
appropriate total entropy and free energy. From this it is clear that the
entropy is simply additive,
$S_{\mathrm{tot}}(X_{A},X_{B})=S_{A}(X_{A})+S_{B}(X_{B})$, but it is the free
energy divided by temperature, rather than the free energy itself, that is
additive,
$F_{\mathrm{tot}}(X_{A},X_{B})/T_{\mathrm{eff}}=F_{A}(X_{A})/T_{A}+F_{B}(X_{B})/T_{B}$.
It is only in the case of subsystems all with the same temperature that the
free energy is extensive (i.e. simply additive). By the usual principles of
mechanics, the energy is simply additive
$E_{\mathrm{tot}}(X_{A},X_{B})=E_{A}(X_{A})+E_{B}(X_{B})$, which result will
prove essential for the following analysis.

In searching for the thermodynamic basis of the H. London expression,HLondon39
Eq. (2.1), there are two possible axioms to be considered. The first is that
the total entropy of the total system is a maximum, which is of course just
the Second Law of Thermodynamics, albeit applied to a non-equilibrium steady
state system.NETDSM  The second is that the total energy of the subsystems be
minimized, which is a mechanical law that is generally without thermodynamic
relevance, the present case being the only significant exception that I can
think of.

A third possible axiom, that the total free energy of the total system is a
minimum is not viable for two related reasons. First, as just explained, it is
not free energy but free energy divided by temperature that is the relevant
thermodynamic potential, so it should be the sum of the free energy divided by
temperature that is a minimum. And second, the free energy divided by
temperature is derived directly from the total entropy, and so the principle
of free energy divided by temperature minimization is not separate to, and
does not yield any new information beyond, total entropy maximization. This
can be seen by the trivial relationship between the second and third
equalities in the above equation. To be clear and unambiguous on this point:
for the case of systems with different temperatures, there is no principle of
free energy minimization. There is a principle of free energy divided by
temperature minimization, but this is no different to the principle of entropy
maximization.

Consider now the first possible axiom, that the total entropy of the system is
a maximum. Since the systems are closed, the total entropy isTDSM

$S_{\mathrm{tot}}=\frac{-F(N_{A},V_{A},T_{A})}{T_{A}}-\frac{F(N_{B},V_{B},T_{B})}{T_{B}},$
(2.3)

where $N$ is the number, $V$ is the volume, and $F$ is the Helmholtz free
energy. (The total subsystem-dependent entropy of a sub-system and its
reservoir is the negative of the free energy divided by the temperature.)TDSM
With the total number of helium atoms fixed, $N=N_{A}+N_{B}$, its derivative
isTDSM

$\frac{\partial S_{\mathrm{tot}}}{\partial
N_{A}}=\frac{-\mu_{A}}{T_{A}}+\frac{\mu_{B}}{T_{B}},$ (2.4)

where $\mu$ is the chemical potential. The maximum total entropy occurs when
this is zero, which gives the condition for the optimum steady state that
follows from the first possible axiom as

$\frac{\mu_{A}}{T_{A}}=\frac{\mu_{B}}{T_{B}}.$ (2.5)

Consider now the second possible axiom, that the total energy is a minimum.
The energy of each chamber is a function of its entropy, volume, and number,
$E_{\mathrm{tot}}=E(S_{A},V_{A},N_{A})+E(S_{B},V_{B},N_{B})$, which is
standard.TDSM  In this case the derivative at fixed $N$ isTDSM

$\frac{\partial E_{\mathrm{tot}}}{\partial N_{A}}=\mu_{A}-\mu_{B}.$ (2.6)

Obviously the minimum energy state corresponds to

$\mu_{A}=\mu_{B}.$ (2.7)

The principle of energy minimization dates to Isaac Newton (1687). It applies
to mechanical systems, in which statistical or entropic considerations are
absent. Although one could obtain this same result by minimizing the simple
sum of the free energies of the two chambers, as discussed above such
procedure is not correct as it is actually the free energy divided by
temperature that is additive.

The chemical potential is the Gibbs free energy per particle,
$\mu=G(N,p,T)/N$.TDSM  The derivative of Eq. (2.7) with respect to $T_{B}$ at
constant pressure and temperature of the first chamber, and number of the
second, isTDSM

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\mathrm{d}(G_{B}/N_{B})}{\mathrm{d}T_{B}}$ (2.8)
$\displaystyle=$ $\displaystyle\frac{\partial g_{B}}{\partial
T_{B}}+\frac{\partial g_{B}}{\partial
p_{B}}\frac{\mathrm{d}p_{B}}{\mathrm{d}T_{B}}$ $\displaystyle=$
$\displaystyle-s_{B}+v_{B}\frac{\mathrm{d}p_{B}}{\mathrm{d}T_{B}},$

where $g$, $s$, and $v=\rho^{-1}$ are the Gibbs free energy, entropy, and
volume per particle, respectively. This is the same as H. London’s expression,
Eq. (2.1). Conversely, the derivative of the expression that results from
maximizing the total entropy, Eq. (2.5), does not yield this result.

Equation (2.7) is the major result of this paper: on the fountain path the
chemical potential is constant, which is the same as saying that the energy is
minimized at constant entropy.

####  II.1.1 Rationalization

A possible way to understand the equality of chemical potential in the
fountain effect is as follows. Each chamber consists of a mixture of ground
state bosons, denoted by subscript zero, and excited state bosons, denoted by
subscript asterisk. These are in equilibrium with each other, and so we can
drop the subscript on the chemical potential, $\mu_{0A}=\mu_{*A}=\mu_{A}$, and
similarly for chamber $B$. This follows because each chamber has uniform
temperature.

We imagine that the transfer of $N$ condensed bosons via the capillary from
chamber $A$ to $B$ is accomplished by a contiguous packet of initial volume
$Nv_{0A}$. This packet has initial pressure, $p_{A}$, initial enthalpy
$h_{0A}$, and the condensed bosons within it have initial chemical potential
$\mu_{0A}$. (The energy per particle and the volume per particle are well-
defined in a mixture and can be calculated by statistical mechanical
techniques.) We suppose that the usual relationships of equilibrium
thermodynamics hold for the packet, including that the chemical potential is
the Gibbs free energy per particle,TDSM

$\mu_{0A}=h_{0A}-T_{0A}s_{0A},$ (2.9)

where $s_{0A}$ is the initial entropy per boson in the packet. A similar
expression holds for $\mu_{0B}$. It is not necessarily assumed that the
initial or final temperatures of the packet equal that of their chamber.

We assume that at the entrance and exit of the capillary the bosons in the
packet are in equilibrium with the bosons in the respective chambers,

$\mu_{0A}=\mu_{A},\mbox{ and }\mu_{0B}=\mu_{B}.$ (2.10)

We assume that no heat flows into the packet during its transit so that the
enthalpy is constant,

$h_{0A}=h_{0B}.$ (2.11)

(The absence of heat flow is the reason that one cannot assume temperature
equality at the termini.)

From these three assumptions it follows that

$\mu_{A}-\mu_{B}=-T_{0A}s_{0A}+T_{0B}s_{0B}.$ (2.12)

If one takes the superfluid condition to be that the entropy of the condensed
bosons is zero, $s_{0A}=s_{0B}=0$, then this says that the chemical potentials
of the two chambers are equal, $\mu_{A}=\mu_{B}$.

Conversely, if the chemical potentials are equal, then the products on the
right hand side must be a constant on the fountain path,

$T_{0A}s_{0A}=T_{0B}s_{0B}=f(\mu_{A}).$ (2.13)

If the internal temperature of the packet is a non-constant function of the
chamber temperature, $T_{0B}(T_{B})$, then because $f(\mu_{A})$ does not vary
with the second chamber temperature, one must have that

$s_{0B}=f(\mu_{A})=0.$ (2.14)

Since the pressure of the packet at its terminus is $p_{B}$, which varies with
$T_{B}$, it is difficult to see how the internal temperature could be
independent of the chamber temperature. Realistically, the only possible value
for a constant internal temperature would be zero. But this cannot be its
value because the enthalpy goes to zero at absolute zero,
$h_{0B}(T_{0B}=0)=0$, at least according to the generally accepted Nernst heat
theoremHLondon39  and measured saturationDonnelly98  data. The vanishing of
the enthalpy and entropy would mean that the chemical potential also vanished,
$\mu_{0B}=h_{0B}+T_{0B}s_{0B}=0$, whereas the equality of chemical potentials
means that in general it is non-zero, $\mu_{0B}=\mu_{B}=\mu_{A}\neq 0$.

###  II.2 Numerical and Experimental Results

In practice in experimental application,Hammel61  the H. London expression for
the derivative of the fountain pressure, Eq. (2.1), is integrated along the
saturation curve,

$p_{B}-p_{A}=\int_{T_{A}}^{T_{B}}\mathrm{d}T^{\prime}\;\rho^{\mathrm{sat}}(T^{\prime})s^{\mathrm{sat}}(T^{\prime}).$
(2.15)

Since the integral for the fountain pressure should be evaluated on the
fountain path rather than the saturation path, one can correct this as
follows. One has, with pressure and temperature as the independent variables,

$\displaystyle\rho s$ $\displaystyle=$
$\displaystyle\rho^{\mathrm{sat}}s^{\mathrm{sat}}+(p-p^{\mathrm{sat}})\left(\frac{\partial(\rho
s)}{\partial p}\right)_{T}^{\mathrm{sat}}$ (2.16) $\displaystyle\approx$
$\displaystyle\rho^{\mathrm{sat}}s^{\mathrm{sat}}+(p-p^{\mathrm{sat}})\rho^{\mathrm{sat}}\left(\frac{1}{T}\frac{\partial
h}{\partial p}-\frac{v}{T}\right)_{T}^{\mathrm{sat}}$ $\displaystyle=$
$\displaystyle\rho^{\mathrm{sat}}s^{\mathrm{sat}}-\alpha^{\mathrm{sat}}(p-p^{\mathrm{sat}}).$

In the second equality the liquid has been taken to be incompressible. Here
the thermal expansivity is $\alpha=-\rho^{-1}{\partial\rho(p,T)}/{\partial
T}$. The final equality follows from the cross-derivative of the Gibbs free
energy TDSM

$\displaystyle\frac{\partial h}{\partial p}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}(\beta G(N,p,T))}{\partial p\partial\beta}$
(2.17) $\displaystyle=$ $\displaystyle\frac{\partial(\beta v)}{\partial\beta}$
$\displaystyle=$ $\displaystyle v-\alpha vT.$

Here and throughout $\beta=1/k_{\mathrm{B}}T$, with $k_{\mathrm{B}}$ being
Boltzmann’s constant. With this fountain pressure integral can be translated
from the fountain path to the saturation path,

$\displaystyle p_{B}-p_{A}$ $\displaystyle=$
$\displaystyle\oint_{T_{A}}^{T_{B}}\mathrm{d}T^{\prime}\;\rho(T^{\prime})s(T^{\prime})$
$\displaystyle\approx$
$\displaystyle\int_{T_{A}}^{T_{B}}\mathrm{d}T^{\prime}\;\Big{\\{}\rho^{\mathrm{sat}}(T^{\prime})s^{\mathrm{sat}}(T^{\prime})$
$\displaystyle\mbox{
}-\alpha^{\mathrm{sat}}(T^{\prime})\big{[}p(T^{\prime};p_{A},T_{A})-p^{\mathrm{sat}}(T^{\prime})\big{]}\Big{\\}}.$

The fountain pressure in the second term in the integrand may be approximated
by that given by the uncorrected expression, Eq. (2.15), rather than iterated
to self-consistency. Numerically using measured data this form is
indistinguishable from the raw H. LondonHLondon39  form on the saturation
path, Eq. (2.15), as is now not shown.

A third form for the fountain pressure will be tested based directly on the
equality of chemical potentials rather than the integral of the fountain
pressure. Writing
$\mu_{B}=\mu_{B}^{\mathrm{sat}}+(p_{B}-p_{B}^{\mathrm{sat}})v_{B}^{\mathrm{sat}}$,
we have

$\displaystyle p_{B}-p_{A}$ $\displaystyle\approx$ $\displaystyle
p_{B}^{\mathrm{sat}}-p_{A}+\rho_{B}^{\mathrm{sat}}(\mu_{B}-\mu_{B}^{\mathrm{sat}})$
(2.19) $\displaystyle=$ $\displaystyle
p_{B}^{\mathrm{sat}}-p_{A}+\rho_{B}^{\mathrm{sat}}(\mu_{A}-\mu_{B}^{\mathrm{sat}}).$

This linear expansion of the exact result must eventually become inaccurate
for large pressure differences. Invariably the experimental measurements are
performed at saturation of chamber $A$, and so all quantities on the right
hand side, including $\mu^{\mathrm{sat}}=h^{\mathrm{sat}}-Ts^{\mathrm{sat}}$,
can be obtained from standard tables.Donnelly98

Figure 1:  Measured and calculated fountain pressure for $T_{A}=$ 1.502 K
(left), 1.724 K (middle), and 1.875 K (right). The symbols are measured
data,Hammel61  the full curve is the saturation line integral form of the H.
London expression, Eq. (2.15),HLondon39  and the dotted curve is for fixed
chemical potential using the incompressible fluid estimate for its departure
from the saturation value, Eq. (2.19). The calculated curves use measured
saturation data from Ref. Donnelly98, .

Figure 1 tests the various equations for the fountain pressure against the
measured values.Hammel61  The calculations use measured data for 4He on the
saturation curve, including the calorimetrically measured entropy.Donnelly98
The chemical potential for Eq. (2.19) was obtained from the measured enthalpy
and entropy.

It can be seen that the H. London expression for the temperature derivative of
the fountain pressure, evaluated as an integral along the fountain curve, Eq.
(2.15), is practically indistinguishable from the measured values. Adding the
correction for the translation from the fountain path to the saturation path,
Eq. (II.2), makes a difference of about $-0.5$% at the highest fountain
pressure shown, which would be indistinguishable from the uncorrected result,
Eq. (2.15), on the scale of the figure.

The equation for the fountain pressure derived directly from the equality of
chemical potential, Eq. (2.19), does not perform as well as the integral form
of the H. LondonHLondon39  expression, particularly at higher fountain
pressures. This is in part due to the fact that a linear expansion has been
used to map from the pressure of the second chamber to the saturation
pressure. And in part it is due to measurement error, which is always
magnified by single point applications and taking the differences between
measured quantities. Integral formulations such as Eq. (2.15) tend to smooth
out random measurement error. That the overestimate of the predicted pressure
for $T_{B}\gtrsim 2$ K is similar in all three cases despite the differences
in fountain pressures of about a factor of three tends to suggest that the
error is not due to the linearization approximation. Rather it appears that
$\mu_{B}^{\mathrm{sat}}$ is systematically too small, which says that either
the measured enthalpy is too small, or else the measured entropy is too large
in this regime. Despite the limited accuracy of Eq. (2.19), the agreement with
the measured data in Fig. 1 is good enough to confirm the thermodynamic
analysis that $\mu_{A}=\mu_{B}$ is mathematically equivalent to
$\mathrm{d}p_{B}/\mathrm{d}T_{B}=\rho_{B}s_{B}$.

##  III Osmotic Pressure in an Incompressible Superfluid

In Appendix B the conventional osmotic pressure for a binary solution is
obtained using classical ideal solution theory. That result may be compared
and contrasted with the present results for a superfluid obtained similarly at
the level of an ideal solution.

Consider a system consisting of two chambers at different temperatures below
the condensation temperature and containing 4He. We give the free energy for a
single chamber, and differentiate it to obtain an expression for the chemical
potential. This allows us to equate chemical potentials and to obtain the
fountain pressure.

The constrained Helmholtz free energy for a single system with $N_{0}$ ground
momentum state bosons and $N_{*}$ excited momentum state bosons is given by
Attard22a ; Attard21

$-\beta
F(N_{0}|N,V,T)=\ln\frac{\Lambda^{-3N_{*}}Q}{N_{*}!V^{N_{0}}}+\sum_{l=2}^{\infty}\frac{N_{*}^{l}}{N^{l-1}}g^{(l)}.$
(3.1)

Here $N=N_{0}+N_{*}$, $\beta\equiv 1/k_{\mathrm{B}}T$, and
$\Lambda\equiv\sqrt{2\pi\beta\hbar^{2}/m}$ is the thermal wavelength. The
series contains the loop grand potentials, which is a quantum effect that
arises from the symmetrization of the wave function. The logarithmic term is
the classical or monomer term, and it contains the configuration integral for
interacting bosons, $Q(N,V,T)$. Compared to classical statistical mechanics of
a binary mixture (see Appendix B), for the condensed boson case there is no
$N_{0}!$ in the denominator, and instead of $\Lambda^{3N_{0}}$ there appears
$V^{N_{0}}$. These differences are a consequence of the non-locality of the
permutations of ground momentum state bosons. Attard22a ; Attard21

We make the incompressible liquid approximation, $V=Nv$. The constrained Gibbs
free energy, $G=F+pV$, for chamber $A$ is then given by

$\displaystyle G(N_{0A}|N_{A},p_{A},T_{A})$ $\displaystyle=$
$\displaystyle\beta_{A}^{-1}\left[N_{*A}\ln\frac{N_{*A}\Lambda_{A}^{3}}{N_{A}v_{A}}-N_{*A}-\ln\frac{Q_{A}}{V^{N_{A}}}\right]$
$\displaystyle\mbox{
}-\sum_{l=2}^{\infty}\frac{\beta_{A}^{-1}N_{*A}^{l}}{N_{A}^{l-1}}g_{A}^{(l)}+p_{A}N_{A}v_{A}.$

Recall that $N_{A}=N_{*A}+N_{0A}$.

Now for an incompressible fluid, since $\mu^{\mathrm{ex}}=p^{\mathrm{ex}}v$
(see Appendix B), the derivative of the configuration integral term vanishes,

$\displaystyle\frac{\partial}{\partial N}\ln\frac{Q(N,V,T)}{V^{N}}$ (3.3)
$\displaystyle=$ $\displaystyle\frac{\partial(-\beta
F^{\mathrm{ex}}(N,V,T))}{\partial N}+v\frac{\partial(-\beta
F^{\mathrm{ex}}(N,V,T))}{\partial V}$ $\displaystyle=$
$\displaystyle-\beta\mu^{\mathrm{ex}}+\beta p^{\mathrm{ex}}v=0.$

This result effectively removes the interactions between the helium-4 atoms,
and makes the analysis more or less the same as that for an ideal gas, apart
from the way in which the fraction of ground and excited momentum state bosons
is calculated or measured. Consequently, inserting the incompressible fluid
assumption at this stage of the analysis is a rather serious approximation. It
would be better to calculate the chemical potential for interacting atoms,
which is quite feasible, and this should be explored in the future.

The derivative of the constrained Gibbs free energy gives the chemical
potential, which with the vanishing of the derivative of the configurational
integral, is

$\displaystyle\mu_{0A}\equiv\left(\frac{\partial G_{A}}{\partial
N_{0A}}\right)_{N_{*A},p_{A},T_{A}}$ (3.4) $\displaystyle=$
$\displaystyle\frac{-\beta_{A}^{-1}N_{A*}v}{V_{A}}+p_{A}v_{A}+\beta_{A}^{-1}\sum_{l=2}^{\infty}(l-1)\frac{N_{*A}^{l}g_{A}^{(l)}}{N_{A}^{l}}$
$\displaystyle=$
$\displaystyle\frac{-f_{*A}}{\beta_{A}}+p_{A}\rho_{A}^{-1}+\beta_{A}^{-1}\sum_{l=2}^{\infty}(l-1)(f_{*A})^{l}g_{A}^{(l)}.$

Here $f_{*A}=N_{*A}/N_{A}$ is the fraction of bosons in excited states in
chamber $A$. One can similarly obtain an expression for the excited state
chemical potential,

$\displaystyle\mu_{*A}$ $\displaystyle=$ $\displaystyle\frac{\partial
G_{A}}{\partial N_{*A}}$ $\displaystyle=$
$\displaystyle\beta_{A}^{-1}\ln\frac{N_{*A}\Lambda_{A}^{3}}{V_{A}}-\beta_{A}^{-1}\frac{N_{*A}v}{V_{A}}+p_{A}v_{A}$
$\displaystyle\mbox{
}+\beta_{A}^{-1}\sum_{l=2}^{\infty}g_{A}^{(l)}\left\\{(l-1)\frac{N_{*A}^{l}}{N_{A}^{l}}-l\frac{N_{*A}^{l-1}}{N_{A}^{l-1}}\right\\}.$

In the equilibrium state, $\mu_{0A}=\mu_{*A}$, in which case one must have

$\ln[f_{*A}\rho_{A}\Lambda_{A}^{3}]=\sum_{l=2}^{\infty}lg_{A}^{(l)}f_{*A}^{l-1}.$
(3.6)

Since the right hand side is a series of positive terms, one must have that
$f_{*A}>1/\rho_{A}\Lambda_{A}^{3}$, which places a lower bound on the fraction
of excited state bosons (He I) as $T\rightarrow 0$. This result is predicated
on the incompressible liquid, ideal solution approximation and also upon the
no mixing approximation.

Similar expressions hold for chamber $B$. Hence equating the chemical
potential of the two chambers, $\mu_{0A}=\mu_{0B}$, yields the fountain
pressure as

$\displaystyle p_{A}-p_{B}$ $\displaystyle=$
$\displaystyle\rho_{A}k_{\mathrm{B}}T_{A}f_{*A}-\rho_{B}k_{\mathrm{B}}T_{B}f_{*B}$
$\displaystyle\mbox{
}-k_{\mathrm{B}}\sum_{l=2}^{\infty}(l-1)\Big{\\{}\rho_{A}T_{A}g_{A}^{(l)}f_{*A}^{l}-\rho_{B}T_{B}g_{B}^{(l)}f_{*B}^{l}\Big{\\}}.$

Ignoring the loop terms, and taking into account the fact that the excited
momentum state fraction increases with increasing temperature, one sees that
the higher temperature chamber has the higher pressure. Again neglecting the
loop terms, notice the difference between this and the usual osmotic pressure
equation (B.4). It is only in the linear regime that they are the same.

The left hand side can be written as a height difference,
$[p_{A}-p_{B}]\rho^{-1}\approx mg[h_{A}-h_{B}]$, where $m$ is the molecular
mass. (Note that here $g$ is the acceleration due to gravity, not the specific
Gibbs free energy, and $h$ is the height, not the specific enthalpy.)

Extensive calculation of the intensive loop Gaussians $g^{(l)}$ up to $l=7$
have been performed for a Lennard-Jones model of 4He,Attard21 ; Attard22a  and
also for 3He.Attard22c  The fraction of excited state bosons following
condensation has been calculated in the so-called unmixed approximation.
Attard22a  These have been used in the present expression for the pressure
difference. The $\lambda$-transition in Lennard-Jones helium-4 occurs at
$T=0.65$. Attard22a

Figure 2:  The height difference as a function of the temperature difference
for fixed $T_{B}=0.65$ (circles) and for fixed $T_{A}=0.50$ (triangles) for
Lennard-Jones 4He, for which the temperature in K is $10.22T$.

Figure 2 shows the calculated height difference as a function of the low
temperature $T_{A}$ for fixed high temperature $T_{B}=0.65$, which is the
$\lambda$\- temperature in Lennard-Jones 4He. It also shows the height
difference for fixed low temperature $T_{A}=0.50$, It can be seen that the
height difference increases with increasing temperature difference. The
magnitude of the fountain effect is surprisingly large considering that the
largest temperature difference shown is about 1.5 K. The loop series
contribution to the total height difference is on the order of 1–5%. This
means that the major part of the height differences in Fig. 2 come from the
first line of the right hand side of Eq. (III), which are essentially ideal
solution terms.

It is noticeable in the figure that for the same temperature difference,
$T_{B}-T_{A}=0.05$, the higher temperature $T_{B}=0.65$ has a height
difference $h_{B}-h_{A}=159.2$ m, which is about 50% larger than that at the
lower temperature $T_{A}=0.50$, which has a height difference
$h_{B}-h_{A}=104.7$ m. This shows that there is a significant decrease in the
ratio of height difference to temperature difference with decreasing
temperature. This is because the fraction of excited state bosons decreases
with decreasing temperature, and to leading order the height difference is
proportional to the difference in the excited fraction.

The experimentally measured value at $T_{A}=1.143$ K is 46.8 m for
$T_{B}-T_{A}=0.6$ K.Hammel61  The calculated height difference in Fig. 2 is
104.7 m for $T_{B}-T_{A}=0.05\approx 0.51$ K, which is about a factor of 2 too
large. The main reason for the disagreement between the calculated and the
measured height difference appears to be the use of the incompressible liquid,
ideal solution approximation for the calculations.

Figure 3:  Measured and calculated fountain pressure for $T_{A}=$ 1.502 K
(left), 1.724 K (middle), and 1.875 K (right). The symbols are measured
data,Hammel61  the full curve is the saturation line integral form of the H.
London expression, Eq. (2.15),HLondon39  and the dotted curve uses the
incompressible fluid osmotic pressure result, the first line of Eq. (III),
using the measured fraction of excited state bosons.Donnelly98  The calculated
curves use measured saturation data from Ref. Donnelly98, .

Figure 3 compares the incompressible fluid osmotic pressure result, the first
line of Eq. (III), with the measured fountain pressure. This uses the
measuredDonnelly98  rather than calculated excited state fraction. It neglects
the loop contributions, which as discussed for the previous figure are
expected to be small. From the fact that the measured curves are nearly
parallel, one can conclude that the fountain pressure is approximately a
function of the temperature difference, $T_{B}-T_{A}$, rather than of $T_{B}$
and $T_{A}$ individually. In any case it can be seen that the incompressible,
ideal solution result for the osmotic pressure performs quite badly.

Figure 4:  MeasuredDonnelly98  fraction of He I times the measuredDonnelly98
number density and thermal wavelength cubed (solid curve). The dotted line is
a guide to the eye.

Figure 4 shows the dimensionless parameter $f_{*}\rho\Lambda^{3}$ using
measured data for the fraction of He I. Donnelly98  As discussed following Eq.
(3.6), equilibrium between ground and excited state bosons implies that this
must be greater than unity, which the figure shows is violated for $T\leq 1.5$
K. The tests of the osmotic pressure prediction for the fountain pressure in
Fig. 3 are for $T\geq 1.5$ K. Nevertheless that $f_{*}\rho\Lambda^{3}<1$ for
$T\leq 1.5$ K is presumably an indication of the failure of the incompressible
liquid, ideal solution, and no mixing approximations in the low temperature
regime.

##  IV Conclusion

The osmotic pressure mechanism offered by TiszaTisza38  as an explanation for
the fountain pressure is problematic on physical grounds. In the usual
realization of osmotic pressure the solute and the solvent have distinct
identities. Hence the only way to equalize the solvent chemical potential is
to increase the pressure of the high concentration chamber. In the case of
4He, the ground state bosons can become excited state bosons, and _vice
versa_. Hence if the different fractions in each chamber somehow corresponded
to a chemical potential difference due to mixing entropy, they could equalize
chemical potential simply by changing their state without changing the
pressure.

The present calculations show that the osmotic pressure mechanism can be an
order of magnitude or more too large for the fountain pressure. Admittedly
this is using the incompressible fluid, ideal solution approximation. But
since the common understanding of osmotic pressure is that it is due to mixing
entropy, and since the latter can be calculated exactly by ideal
combinatorics, the failure of the present ideal solution calculations for the
fountain pressure argue against osmotic pressure as the physical basis of the
fountain effect.

It was found here that H. London’sHLondon39  expression for the temperature
derivative of the fountain pressure corresponds to chemical potential equality
of the two chambers. This follows from the minimization of energy at constant
entropy, which suggests a general principle for superfluid flow and offers an
alternative physical explanation for the fountain effect.

Imagine that the high temperature chamber $B$ is initially at saturation. For
4He, on the saturation curve the chemical potential from the
measuredDonnelly98  enthalpy and entropy decreases with increasing
temperature, $\mathrm{d}\mu^{\mathrm{sat}}/\mathrm{d}T<0$. Hence
$\mu_{B}^{\mathrm{sat}}<\mu_{A}^{\mathrm{sat}}$. Since $\partial\mu/\partial
p=v>0$, the chemical potential of the high temperature chamber can be
increased to achieve equality by increasing the pressure beyond its saturation
value. This occurs as more bosons arrive in the second chamber because each
boson occupies a certain impenetrable volume (i.e. the compressibility is
positive). These are the reasons why condensed bosons initially flow down the
chemical potential gradient from the low temperature chamber to the high
temperature chamber, and why the high temperature chamber subsequently settles
at a higher pressure.

One could describe the new principle —that the energy at constant entropy is
minimal with respect to particle transfer— as an empirical formula, since the
final formula agrees quantitatively with measured results but the axiomatic
basis for energy minimization is not well-established. Undoubtedly one could
argue that condensed bosons in the superfluid state are in a state of low or
zero entropy, and so it would make sense that they respond to mechanical
forces rather than to statistical or entropic forces. Also the fact that
superfluid flow is flow without viscous dissipation means that the motion of
condensed bosons does not produce entropy and so it cannot respond to entropy
gradients. And of course condensed bosons in a single quantum state behave as
a single particle,Attard21 ; Attard22a  which particle responds as a whole to
mechanical forces without changing the entropy of the system. Perhaps time and
familiarity will raise the status of the present result from empirical to
axiomatic.

It is unlikely that the principle of energy minimization at constant entropy
is restricted to the fountain effect. Presumably it is a fundamental principle
that applies generally for motion without dissipation in Bose-Einstein
condensates. As such it provides new insight for the understanding and the
analysis of superfluid flow and of superconductor currents.

## References

  * (1) J. F. Allen and H. Jones, “New Phenomena Connected with Heat Flow in Helium II”, _Nature_ 141, 243 (1938).
  * (2) S. Balibar “Laszlo Tisza and the Two-Fluid Model of Superfluidity”, _C. R. Physique_ 18, 586 (2017).
  * (3) R. J. Donnelly and C. F. Barenghi, “The Observed Properties of Liquid Helium at the Saturated Vapor Pressure”, _J. Phys. and Chem. Ref. Data_ 27, 1217 (1998).
  * (4) L. Tisza, “Transport Phenomena in Helium II”, _Nature_ 141, 913 (1938).
  * (5) F. London, “The $\lambda$-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy”, _Nature_ 141, 643 (1938).
  * (6) H. London, “Thermodynamics of the Thermomechanical Effect of Liquid He II” _Proc. Roy. Soc._ A171, 484 (1939).
  * (7) E. F. Hammel, Jr. and W. E. Keller, “Fountain Pressure Measurements in Liquid He II”, _Phys. Rev._ 124, 1641 (1961).
  * (8) P. Attard, _Quantum Statistical Mechanics in Classical Phase Space_ (IOP Publishing, Bristol, 2021).
  * (9) P. Attard, “Bose-Einstein Condensation, the Lambda Transition, and Superfluidity for Interacting Bosons”, arXiv:2201.07382 (2022).
  * (10) P. Attard, “Attraction Between Electron Pairs in High Temperature Superconductors”, arXiv:2203.02598 (2022).
  * (11) P. Attard, “New Theory for Cooper Pair Formation and Superconductivity”, arXiv:2203.12103v2 (2022).
  * (12) P. Attard, _Thermodynamics and Statistical Mechanics: Equilibrium by Entropy Maximisation_ (Academic Press, London, 2002).
  * (13) P. Attard, _Non-Equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications_ (Oxford University Press, Oxford, 2012).

##  Appendix A Critique of H. London’s Derivation

As part of the preparation for this paper, a detailed study of H. London’s
derivationHLondon39  of his expression was undertaken, and the conclusions are
reported here. The criticisms of the derivation are rather serious and suggest
a disconnect with the final result. The reader may be sceptical on this point,
given the overwhelming experimental evidence for the quantitative accuracy of
the H. London expression. And in any case some may wonder whether details of
the derivation matter given that the final expression works. But it seems to
me that the mathematical derivation does matter because it establishes the
status of the final expression: is it a formally exact thermodynamic result,
or is it only an accurate approximation? If a theoretical expression is to be
used to calibrate instruments, or as a benchmark against which to judge
different experimental techniques, then it must be provably exact, not merely
an approximation accurate to a limited extent over a limited regime.
Establishing the precise status of the H. London expression is particularly
important in the case of liquid helium where measured data are reported to
four or even more significant figures. Donnelly98

The details of the derivation of H. LondonHLondon39  also matter because they
set a precedence for proper thermodynamic manipulation, both in general and in
the particular case of superfluid flow. The particular combination of
artificial modeling, neglected terms, and implicit assumptions may well give
the correct answer on this occasion, but not more generally. Of course since
quantitative values of fountain pressure measurements were known to H. London
at the time,HLondon39  the fact that his final expression fits the measured
data provides little comfort that his derivation of that expression is sound.
Presumably to some extent he worked backwards, trying different models and
neglecting different terms until he got a result that fitted the known data.

###  A.1 Artificial Model

Figure 5:  Model system analyzed by H. London.HLondon39

The model used by H. LondonHLondon39  for his derivation of the fountain
pressure expression is shown in Fig. 5. In the first half of the cycle, with
the valves to the engine closed, the pistons on the chambers are used to
transfer $N$ bosons from $A$ to $B$ via the capillary. In the second half of
the cycle, sequentially operating the valves and the pistons, the $N$ bosons
are transferred back from $B$ to $A$ via the engine following the fountain
path $p^{\prime}=p(T^{\prime};p_{A},T_{A})$. Heat flows in to or out of the
helium and work is done on or by it at each stage.

The first criticism of H. London’s derivationHLondon39  is the artificiality
of this model. What has this got to do with the actual experimental
arrangement for measuring the fountain pressure? Are the various heat flows
that occur in the artificial engine the same as those that occur in the actual
measurement setup? What exactly is superfluid about the model?

My second criticism is that this artificial model seems designed to avoid
using an appropriate thermodynamic potential for the system, such as the
Helmholtz or Gibbs free energy. These free energies are entirely absent from
H. London’s paper.HLondon39  Also absent is the chemical potential, which is
the usual thermodynamic variable involved in number exchange.

My third criticism is that the first mention of a specific superfluid property
occurs after Eq. (10) of the paper by H. London,HLondon39  which is
interpreted to mean that the superfluid has zero entropy. This is after the
derivation has been completed; Eq. (11) is the expression for the fountain
pressure derivative, the present Eq. (2.1). In neither the model itself or the
equations used for the derivation is any restriction or special form due to
superfluidity imposed. In other words, the result of the derivation should
apply equally to an ordinary fluid. The only way that the correct fountain
pressure derivative for a superfluid can emerge from the derivation is by some
fortuitous error, or implicit assumption, or neglected contribution.

It is true that early in the derivation H. LondonHLondon39  states that he is
treating reversible and irreversible effects as separable and independent, and
that he is neglecting the latter. But this is not a property confined to
superfluids, since it is a common assumption in irreversible thermodynamics,
and it underpins, for example, the formulation of linear hydrodynamics. NETDSM

###  A.2 Thermodynamic and Mathematical Details

My fourth and final criticism is of the mathematical derivation itself. There
are three key equations: the work done by the helium, the change in energy in
a cycle, and the change in entropy in a cycle.

H. LondonHLondon39  gives the total work done by the helium in a cycle as his
Eq. (1a)

$W=N\oint_{T_{A}}^{T_{B}}\mathrm{d}T^{\prime}\;v^{\prime}\frac{\mathrm{d}p^{\prime}}{\mathrm{d}T^{\prime}}.$
(A.1)

The integral is on the fountain path, $p^{\prime}=p(T^{\prime};p_{A},T_{A})$.
This equation is correct.

H. LondonHLondon39  defines the heat flow into the helium in chamber $A$ from
the reservoir as $Q_{A}$, and similarly for chamber $B$. These neglect any
energy change via the capillary.

H. LondonHLondon39  in his Eq. (2) gives the energy conservation equation,
otherwise known as the First Law of Thermodynamics, for a full cycle

$Q_{A}+Q_{B}-N\oint_{T_{A}}^{T_{B}}\mathrm{d}T^{\prime}\;c^{\mathrm{s}}_{p_{A},T_{A}}(T^{\prime})=N\oint_{T_{A}}^{T_{B}}\mathrm{d}T^{\prime}\;v^{\prime}\frac{\mathrm{d}p^{\prime}}{\mathrm{d}T^{\prime}}.$
(A.2)

Here
$c^{\mathrm{s}}_{p_{A},T_{A}}(T^{\prime})=T^{\prime}\mathrm{d}s(T^{\prime};p_{A},T_{A})/\mathrm{d}T^{\prime}$
is the heat capacity per particle along the fountain path (my notation). Here
I’ve neglected the thermal conductivity in the capillary,
$\mu_{\mathrm{cond}}=0$, as H. London does later in his analysis.HLondon39
The third term on the left hand side is the heat flow into the helium in the
engine. Since the energy change in the helium in a full cycle is zero, the
heat flow into the helium must equal the work done by the helium. This
equation is correct.

In actual fact the integrals can be performed to give

$Q_{A}+Q_{B}+Nh_{A}-Nh_{B}=0,$ (A.3)

where $h$ is the enthalpy per particle. It is a little surprising that H.
LondonHLondon39  did not avert to this.

H. LondonHLondon39  in his Eq. (3) gives his version of the Second Law of
Thermodynamics for the cycle,

$\frac{Q_{A}}{T_{A}}+\frac{Q_{B}}{T_{B}}-N\oint_{T_{A}}^{T_{B}}\mathrm{d}T^{\prime}\;\frac{c^{\mathrm{s}}_{p_{A},T_{A}}(T^{\prime})}{T^{\prime}}=0.$
(A.4)

In referring to this equation as the Second Law of Thermodynamics, H.
LondonHLondon39  does not appear to be making any statement about entropy
maximization. Rather he seems to be making reference to the fact that the heat
flow divided by temperature gives a change in entropy.

H. LondonHLondon39  appears to believe that this is an equation for the change
in entropy of the subsystems (the chambers) alone, which by definition must be
zero for a full cycle. But this equation is not the true equation for the
change in entropy of the subsystems as it neglects the work done. Obviously
any $pV$-work on a system changes its energy and therefore its entropy; indeed
at constant number, $\mathrm{d}S=\mathrm{d}E/T+p\mathrm{d}V/T$. The
contribution of the work done by the helium in the chambers in changing their
volume is neglected in H. London’sHLondon39  Eq. (3).

Also there is no conservation law for entropy. It is one thing to say that the
conserved variables such as energy return to the same state after a full
cycle, and that therefore flows in them have to add to zero; it is quite
another thing to attempt to analyze flows in non-conserved variables such as
entropy. I don’t see the point in attempting to analyze the change in entropy
in a cycle unless one intends to maximize it, in which case it should be the
total entropy of the chambers and the thermal reservoirs, not just of the
chambers.

I also point out the very specific model used by H. LondonHLondon39  for
performing work on the helium in the chambers. Such mechanical pistons do not
change the entropy of the reservoirs external to the chambers. If one used a
more realistic model for the external work, such as the Gibs free energy to
represent fluid external to that part of the chamber that is the focus, then
one would have to take into account the change in external entropy as work was
done on the helium in the chamber. In this case one does not arrive at H.
London’sHLondon39  result for the temperature derivative of the fountain
pressure.

In summary, whilst I do not doubt the validity of the expression for the
temperature derivative of the fountain pressure given by H. London,HLondon39
the present Eq. (2.1), I have grave misgivings about the derivation he offers
for it. A theory has to be convincing on its own mathematical and axiomatic
merits. A theory cannot be justified by agreement with experiment when the
process of developing the end product consists of tinkering with the
derivation and model until a fitting result is obtained. Is the successful
‘best fit’ selected for publication any more convincing than those models that
didn’t make it?

In the case of H. London’s formula,HLondon39  I find his derivation flawed on
several grounds. It appears to me that the combination of the artificial
cycle, the unrealistic mechanical external work, the accident that the
incorrect expression for the change in entropy of the chambers actually equals
the correct expression for the change in the thermal reservoirs, all conspire
to give the known experimental result for the fountain pressure even though no
superfluid property is actually input into the derivation. Because of these
flaws in the derivation, the exact status of the H. LondonHLondon39  formula
for the fountain pressure is in my opinion unproven.

##  Appendix B Incompressible Classical Ideal Gas

Consider two chambers, $A$ and $B$, connected by a semi-permeable membrane
transparent to the solvent 1. Assume that the solvent and solute are
incompressible, so that the the volume of the first chamber is
$V_{A}=N_{A1}v_{1}+N_{A2}v_{2}$, and similarly for the second chamber. The
total number is fixed, $N_{n}=N_{An}+N_{Bn}$, $n=1,2$. For the classical ideal
gas, the constrained Gibbs free energy is given byTDSM

$\displaystyle\beta G(N_{A1}|\underline{N},p_{A},p_{B},T)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{2}\Big{\\{}N_{An}\ln\frac{N_{An}\Lambda^{3}}{N_{A1}v_{1}+N_{A2}v_{2}}-N_{An}$
$\displaystyle\mbox{
}+N_{Bn}\ln\frac{N_{Bn}\Lambda^{3}}{N_{B1}v_{1}+N_{B2}v_{2}}-N_{Bn}\Big{\\}}$
$\displaystyle\mbox{ }+\beta p_{A}[N_{A1}v_{1}+N_{A2}v_{2}]+\beta
p_{B}[N_{B1}v_{1}+N_{B2}v_{2}].$

Here $\beta=1/k_{\mathrm{B}}T$ is the reciprocal temperature,
$\Lambda=\sqrt{2\pi\hbar^{2}\beta/m}$ is the thermal wavelength, $m$ is the
mass, and $\mu$ is the chemical potential. Since $G=\mu N$, this shows that
for an incompressible fluid, $\mu^{\mathrm{ex}}=p^{\mathrm{ex}}v$. The
derivative is

$\displaystyle\left(\frac{\partial\beta G}{\partial N_{A1}}\right)_{N_{1}}$
$\displaystyle=$
$\displaystyle\ln\frac{N_{A1}\Lambda^{3}}{N_{A1}v_{1}+N_{A2}v_{2}}-\ln\frac{N_{B1}\Lambda^{3}}{N_{B1}v_{1}+N_{B2}v_{2}}$
(B.2) $\displaystyle\mbox{
}-\frac{N_{A1}v_{1}}{N_{A1}v_{1}+N_{A2}v_{2}}+\frac{N_{B1}v_{1}}{N_{B1}v_{1}+N_{B2}v_{2}}$
$\displaystyle\mbox{
}-\frac{N_{A2}v_{1}}{N_{A1}v_{1}+N_{A2}v_{2}}+\frac{N_{B2}v_{1}}{N_{B1}v_{1}+N_{B2}v_{2}}$
$\displaystyle\mbox{ }+\beta p_{A}v_{1}-p_{B}\beta v_{1}$ $\displaystyle=$
$\displaystyle\ln\frac{\rho_{A1}}{\rho_{B1}}-[\rho_{A1}+\rho_{A2}-\rho_{B1}-\rho_{B2}]v_{1}$
$\displaystyle\mbox{ }+\beta[p_{A}-p_{B}]v_{1}.$

Hence at equilibrium

$\ln\frac{\overline{\rho}_{A1}}{\overline{\rho}_{B1}}-[\overline{\rho}_{A1}+\overline{\rho}_{A2}-\overline{\rho}_{B1}-\overline{\rho}_{B2}]v_{1}v_{1}=-\beta[p_{A}-p_{B}]v_{1}.$
(B.3)

For simplicity choose $v_{1}=v_{2}=v$, so that $\rho=\rho_{1}+\rho_{2}=v^{-1}$
and $\rho_{A}=\rho_{B}$. In this case the equilibrium condition becomes

$\ln\frac{\overline{\rho}_{A1}}{\overline{\rho}_{B1}}=\frac{-\beta}{\rho}[p_{A}-p_{B}].$
(B.4)

Hence $p_{A}\geq p_{B}$ implies that
$\overline{\rho}_{A1}\leq\overline{\rho}_{B1}$. The chamber with the greater
pressure is the one with the lower solvent fraction (and hence higher solute
fraction). This is the classical osmotic pressure.

##  Appendix C Irreversible Heat Flow

The body of the paper has been concerned with the reversible contributions to
the fountain effect, which determine the structure and pressures of the two
chambers. It is also of interest to explore at an elementary level of
approximation the irreversible contributions, which determine the heat flow in
the steady state.

Assuming the enthalpy of the packet is constant, then for $N$ bosons making
the passage $A\rightarrow B$ via the capillary the heat flow into the helium
is

$\displaystyle Q_{B}^{\mathrm{irrev}}$ $\displaystyle\approx$ $\displaystyle
Nu_{B}-[Nu_{A}-(p_{B}-p_{A})Nv].$ (C.1)

This assumes that changes in the volume per boson can be neglected. Assuming
that the initial energy per particle of the packet is representative of the
chamber as a whole, $u_{0A}\approx u_{A}$, allows the irreversible heat flow
in the starting chamber to be taken to be zero, $Q_{A}^{\mathrm{irrev}}=0$.
Obviously these approximations could be refined.

For the return passage $B\rightarrow A$ via the capillary, the irreversible
heat flow is

$\displaystyle Q_{A}^{\mathrm{irrev}}$ $\displaystyle\approx$ $\displaystyle
Nu_{A}-[Nu_{B}+(p_{B}-p_{A})Nv].$ (C.2)

Again the starting energy is taken to be representative, $u_{0B}\approx
u_{B}$, and the irreversible heat flow in the starting chamber is set to zero,
$Q_{B}^{\mathrm{irrev}}=0$.

Now the change in entropy of the universe upon cycling a packet of $N$ bosons
between the chambers is

$\displaystyle\frac{\Delta^{\mathrm{irrev}}S_{\mathrm{tot}}}{N}$
$\displaystyle=$
$\displaystyle\frac{-Q_{B}^{\mathrm{irrev}}}{NT_{B}}+\frac{-Q_{A}^{\mathrm{irrev}}}{NT_{A}}$
$\displaystyle\approx$
$\displaystyle[u_{B}-u_{A}+(p_{B}-p_{A})v]\frac{T_{B}-T_{A}}{T_{A}T_{B}}.$

Since both the energy and the pressure increase with temperature, this is
positive and quadratic in the temperature difference, as it must be. In the
steady state, this drives simultaneous forward and backward superfluid flow in
the capillary.