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superfluid helium-4 is a typical boson fluid to occur in nature and bose condensation is believed to be the fundamental reason for such a phenomenon . recently ,
superfluidity of liquid helium-4 in highly connected porous structures ( vycor glass , glass plate , xerogel , aerogel , graphite , fine powders , steel , german silver , plastic films , _
etc_. ) has been observed and studied intensively .
@xcite however , so far no satisfactory explanation of experimental observations on these materials has been achieved .
@xcite since porous media can be treated as solids with fractal or non - integer dimensions , @xcite we believe that it is imperative to study the extent of the dimensionality contribution to the superfluidity in order to understand the experimental results .    in this paper
, we examine the physical properties of ideal bose gas with fractal dimensions between @xmath0 ( thin film limit ) and @xmath1 ( the bulk limit ) .
the results will be compared with experimental results obtained from liquid helium-4 in porous media .
surprisingly , we find that most of salient features of experimental observations on liquid helium-4 in porous solids can be explained using the theoretical results from the ideal gas model in fractal dimensions .
this indicates that the dimensionality contribution , not the mutual interactions , is the dominant factor for superfluidity in porous media .
the ideal bose gas system at integer dimensions was studied long time ago.@xcite however , study at fractal dimensions was carried out only quite recently.@xcite the density of states which is essential to calculate thermodynamic properties of the bose gas in @xmath2-dimension , where @xmath2 is any real number , is given by @xcite @xmath3 where @xmath4 is the @xmath2-dimensional coefficient which is known as @xmath5 here , @xmath6 is the gamma function , @xmath7 the mass , and @xmath8 the @xmath2-dimensional _ measure _ or volume .    in order to obtain physical quantities of the @xmath2-dimensional bose gas , it is necessary to obtain the grand partition function in @xmath2-dimension , @xmath9 where @xmath10 is the thermodynamic fugacity defined by @xmath11 , @xmath12 , and @xmath13 is the thermal wavelength defined by @xmath14 . with the coefficient @xmath4 , we can readily calculate the grand partition function in @xmath2-dimension .
the @xmath15 is the bose gas function defined by @xmath16 the coefficients @xmath17 and @xmath10 are restricted to @xmath18 and @xmath19 regions .
the bose gas function can be also extended to non - integer dimensions,@xcite and has an integral expression , @xmath20 note that this representation is valid only when @xmath21 .    using the above expressions
, we obtain the average number of particles in @xmath2-dimension , @xmath22 we use the above equations to calculate thermodynamic properties in fractal dimensions in the following .
before we go into detailed calculations , we briefly summarize experimental findings on superfluid helium-4 in porous media , since we are interested in whether the dimensionality considered above plays any role in real situations .
it is known that physical properties of superfluid helium-4 in porous media show quite puzzling behaviors . @xcite upon reducing the lab variables of porous media such as  pore filling , "  thickness , "  number of layers , " or 
coverage , " the capacity curves show the following generic behaviors independent of the porous media or substrates : ( i ) the critical temperature for superfluidity onset shifts downwards almost linearly .
( ii ) the bulk - like sharp cusp disappears and the peak gets smaller .
furthermore , there appears a systematic crossing at low temperatures .
( iii ) the shape of the curve gets rounder and , eventually , becomes flat .
these experimental findings are summarized in fig .
these figures are reproduced from references [ 1 - 4 ] , so that comparison with the theory can be made transparent . here
, we observe that the above experimental findings appear independent of detailed nature of porous media and also of interactions between particles .
this fact strongly suggests that , at least , the qualitative nature of the above behavior may originate from geometric factors .
( iv ) another interesting physical property is the superfluid density shown in fig .
2 . the sample e has smaller density , which implies more connectivity of porous structure than the sample
it is shown that the curves from different media do not cross one another .
this again indicates that the superfluid density is strongly dependent on the porous structure .
numerous theories have been suggested to explain the above experimental observations .
@xcite however , so far no completely successful theory has emerged .
for example , the kt model of the vortex mediated transition in two - dimensional space gives an excellent explanation of question ( i ) for very thin films of one or two layers but not for other samples .
we now calculate the physical properties in fractal dimensions and compare the results with the experimental findings summarized above .
fractal dimensionality measures disorderness in terms of the connectivity of the system .
\(i ) the critical temperature , @xmath23 , for the superfluidity of the ideal bose gas in fractal space can readily obtained from eq .
( [ anp ] ) to be given by @xcite @xmath24^{\frac{2}{d}}}. \label{ctem}\ ] ] the almost linear behavior of @xmath23 as a function of dimensionality is plotted in fig .
necessary parameters are taken from reference 25 .
the formula can be simplified when @xmath2 approaches to 2 to be given by @xcite @xmath25 the results are surprisingly in good agreement with the experimental findings shown in fig .
1 . even though we completely neglected the interactions between bose particles , the @xmath26 limit has the same form with the kt theory prediction.@xcite note that the kt theory can not account the experimental results for thick films , but the present @xmath27 limit agrees with the correct bulk value .
\(ii ) the specific heat of the ideal bose gas in fractal space can readily obtained , too , from the grand partition function @xmath28 in eq .
( [ 23 ] ) .
( a ) when @xmath29 @xmath30 and , ( b ) when @xmath31 @xmath32 the @xmath33 curves for several values of fractal dimensions are plotted as functions of temperatures in fig .
we observe that the cusp disappears when the dimension is less than 3 , and the peak height becomes smaller with decreasing dimensionality .
also , fig .
4 shows that there exists a systematic crossover at low temperature regions .
these results are in excellent agreement with the experiments shown in fig
\(iii ) in fig .
4 , we observe that the peaks of the specific heat curves become less and less prominent and , eventually , the curves become flat with decreasing dimensionality .
we show that this behavior originates from a hidden hierarchy in the superfluidity transition with fractal dimensions .
the first temperature derivative of the specific heat can be readily obtained from eqs .
( [ cvl ] ) and ( [ cvh ] ) .
( a ) when @xmath29 @xmath34 ( b ) when @xmath31 @xmath35 ^ 2 } + \frac{d}{2}\frac{g_{d/2}(z ) g_{d/2 - 2}(z)}{[g_{d/2 - 1}(z)]^2 } \right\}. \label{ccvh}\end{aligned}\ ] ] the first derivative of @xmath36 is plotted in fig . 5 in arbitrary unit .
the curves again show the shift of @xmath23 with decreasing dimensionality .
also , it is shown that the discontinuity of @xmath37 disappears with decreasing dimensionality . in order to investigate this behavior
more closely , we studied the relation between the continuity of higher derivatives of @xmath36 and the fractional dimensionality .
taking higher derivatives on @xmath36 and considering the behavior at @xmath23 , we obtain the following relation : @xmath38 = \lim_{\eta \rightarrow 0 }     \sum_{j=1}^n a_{nj } \ , \eta ^{j+2-\frac{(j+1)d}{2 } } ,   \label{crt}\ ] ] where the coefficients @xmath39 are finite constants , and @xmath40 and @xmath41 are the specific heats below and above @xmath23 .
this formula can also be proved by mathematical induction method ( see appendix ) .
the hierarchy of the superfluidity transition with the fractal dimensionality obtained from the above formula is summarized at table 1 .
this table explains the physical origin of the roundness tendency of the specific heat curves with decreasing the dimensionality .
\(iv ) in real system the superfluid fraction is different from the condensate fraction , but the structure of the condensate fraction may shed some clue for the superfluid fraction .
the condensate fraction in @xmath2-dimension can readily be obtained from eqs .
( [ anp ] ) and ( [ ctem ] ) to be given by @xmath42^{\frac{d}{2 } } \label{41}\ ] ] the curves are shown for several fractal dimensions in fig .
it is shown that the curves with higher dimensions have higher @xmath23 and the curves do not cross each other . the fig .
6 gives the information that the sample e has a higher fractal dimension than the sample f in fig .
also , the aerogel has the highest fractal dimension , the xerogel is the next , and the vycor has the lowest one in fig .
of course , they are all less than the bulk value of @xmath43 .
the qualitative features of the theoretical curves are in excellent agreement with the experimental results in fig .
even the tail structure at @xmath44 is reproduced .
physical properties of the ideal bose gas in fractal dimensions are studied theoretically .
the results are compared with the experimental findings from the superfluid helium-4 in porous media .
it is found that the main characteristics of the experimental results are in excellent agreement with the theoretical results obtained from the simple ideal bose gas in fractal dimensions . such a good agreement may not be totally unexpected .
careful examination of the experimental results reveals that the salient features are independent of materials and mutual interactions , thus suggesting that the dominant contributions should come from geometrical factors .
another important point of the theoretical study is the existence of a hierarchy of the superfluidity transitions with changing fractal dimensionality .
the peak of heat capacity is not necessary for the superfluid transition in porous media
. it will be interesting to study how the such physical properties modified when interactions are included in the calculations .    the author thanks to p. pfeifer and c.e .
campbell for useful discussions .
this research was partially supported by the mokpo national maritime university and the korean research foundation ( 98 - 015-d00061 ) .      here ,
we prove eq .
( [ crt ] ) using the mathematical induction method .
first , we give some useful relations of the bose gas function which are needed for the derivation : @xmath45 where @xmath46 is the riemann - zeta function .
@xmath47 we can use above relations for @xmath48 , too .
we put @xmath49 , then @xmath50 as @xmath51 .
we introduce a differential operator defined by @xmath52 where @xmath40 and @xmath41 are the specific heat below and above @xmath23 . for convenience
, we drop the limit notation of ` @xmath53 ( or @xmath54 ) ' during the proof .
\(ii ) we assume that eq .
( [ crt ] ) is true for any positive integer @xmath57
. then @xmath58 using eqs .
( [ sim2 ] ) and ( [ 210 ] ) , we obtain @xmath59 the @xmath60 satisfies the recurrence relation @xmath61 where @xmath62      h.p.r .
frederikse , physica , * 15 * , 860 ( 1949 ) .
brewer , j. low temp .
phys . * 3 * , 205 ( 1970 ) .
m. bretz , phys .
lett . * 31 * , 1447 ( 1973 ) .
d. finotello , k.a .
gillis , a. wong , and m.h.w .
chan , phys .
lett . * 61 * ( 1988 ) .
l.m . steele and d. finotello ,
j. low temp .
phys . * 89 * , 645 ( 1992 ) .
reppy , _ phase transitions in surface films _ , edited by j.g .
dash and j. ruvalds ( plenum , new york , 1980 ) p. 233 .
reppy , j. low temp . phys . *
87 * , 205 ( 1992 ) .
wong , p.a .
crowell , h.a .
cho , and j. d. reppy , phys .
b. * 48 * , 3858 ( 1993 ) .
bowley and s. giorgini , j. low temp .
93 * , 987 ( 1993 ) .
a. khurana , physics today , * 42 * , 21 ( 1989 ) .
goble and l.e.h .
trainor , can .
* 44 * , 27 ( 1966 ) .
kosterlitz and d.j .
thouless , j. phys .
c * 6 * , 1181 ( 1973 ) .
crowell , f.w .
van keuls , j.d .
reppy , phys .
b. * 55 * , 12620 ( 1997 ) .
phillips , d. ross , p. taborek , and j.e .
rutledge , phys .
b. * 58 * , 3361 ( 1998 ) .
f.m . gaspirini and s. mhlanga , phys
b. * 33 * , 5066 ( 1986 ) .
williams , phys .
lett . * 64 * , 978 ( 1990 ) .
m. saarela , b.e .
clements , e. krotscheck , and f.v .
kusmartsev , j. low temp . phys . * 93 * , 971 ( 1993 ) .
p. pfeifer and m. obert , _ the fractal approach to heterogeneous chemistry _ , edited by d. avnir ( wiely , chichester , 1989 ) p. 11 .
p. pfeifer , y.j .
cole , and j. krim , phys .
lett . * 62 * , 1977 ( 1989 ) .
p. pfeifer and m.w .
cole , new j. chem .
* 14 * , 221 ( 1990 ) .
p. pfeifer , _ chemistry and physics of solid surfaces _ , edited by r. vanselow and r. howe ( springer - verlag , berlin , 1988 ) vol .
vii , p. 283 .
ziff , g.e .
uhlenbeck , and m. kac , phys . rep . *
32 * , 169 ( north - holland , amsterdam , 1977 ) .
kim , int .
* 7 * , 1053 ( 1997 ) .
robinson , phys . rev . * 83 * , 678 ( 1951 ) .
( mcgraw - hill , new york , 1933 ) i-102 .
i.s . gradshteyn and i.m .
ryzhik , _ tables of integrals , series and products _ ( academic , new york , 1980 ) p. 946 .
physical properties of ideal bose gas with the fractal dimensionality between @xmath0 and @xmath1 are theoretically investigated . 
 calculation shows that the characteristic features of the specific heat and the superfluid density of ideal bose gas in fractal dimensions are strikingly similar to those of superfluid helium-4 in porous media . 
 this result indicates that the geometrical factor is dominant over mutual interactions in determining physical properties of helium-4 in porous media .