Appendix: Effects of Correlation between Collisional Stress and

Chapter IV: Mechanical Theory of Phase Coexistence

4.8 Appendix: Effects of Correlation between Collisional Stress and

which is not satisfied in [60]. Indeed, eq. (4.99) shows why∫

𝑑𝑛1/𝑈instead ofΠ^𝑐 could have been used in [60] as an integrating factor to determine the binodal within our framework; the correct integrating factorΠ^𝑐can be expressed as the integral of 1/𝑈under aforementioned Speck’s assumption.

4.8 Appendix: Effects of Correlation between Collisional Stress and Orien-

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⇧

<latexit sha1_base64="QbQ5TBnV0E8NlWPzajt/c3eyKLs=">AAACCXicbZDLSsNAFIZPvNZ6i7p0M1gEQSiJiroRCm50V8FeoI1lMp22Q2eSMDNRSsjWja/ixoUibn0Dd76N0yYLbf1h4OM/53Dm/H7EmdKO823NzS8sLi0XVoqra+sbm/bWdl2FsSS0RkIeyqaPFeUsoDXNNKfNSFIsfE4b/vByXG/cU6lYGNzqUUQ9gfsB6zGCtbE6NmpX2V2CiU7RRcYkRYcZqQcm0o5dcsrORGgW3BxKkKvasb/a3ZDEggaacKxUy3Ui7SVYakY4TYvtWNEIkyHu05bBAAuqvGRySYr2jdNFvVCaF2g0cX9PJFgoNRK+6RRYD9R0bWz+V2vFunfuJSyIYk0Dki3qxRzpEI1jQV0mKdF8ZAATycxfERlgaYIx4RVNCO70ybNQPyq7p+Xjm5NS5TqPowC7sAcH4MIZVOAKqlADAo/wDK/wZj1ZL9a79ZG1zln5zA78kfX5A02KmX4=</latexit>

⇧

act

= ⇧

+ ⇧

swim <latexit sha1_base64="OpcP5lbmFa08FPNtkUyrpd7IItc=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomKeix40VsL9gPaUDbbSbt2swm7G6GE/gIvHhTx6k/y5r9x2+agrQ8GHu/NMDMvSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hoduq3nlBpHssHM07Qj+hA8pAzaqxUd3ulsltxZyDLxMtJGXLUeqWvbj9maYTSMEG17nhuYvyMKsOZwEmxm2pMKBvRAXYslTRC7WezQyfk1Cp9EsbKljRkpv6eyGik9TgKbGdEzVAvelPxP6+TmvDGz7hMUoOSzReFqSAmJtOvSZ8rZEaMLaFMcXsrYUOqKDM2m6INwVt8eZk0zyveVeWiflmu3udxFOAYTuAMPLiGKtxBDRrAAOEZXuHNeXRenHfnY9664uQzR/AHzucPfdCMww==</latexit>

<latexit sha1_base64="FmUnEGw60hHUUQ5NWE26JokDvYs=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARXJUZFXVZcKO7CvYBnaFk0kwbmklCkpGWsQt/xY0LRdz6G+78G9N2Ftp64MLhnHu5955IMqqN5307haXlldW14nppY3Nre8fd3WtokSpM6lgwoVoR0oRRTuqGGkZaUhGURIw0o8H1xG8+EKWp4PdmJEmYoB6nMcXIWKnjHgRJJIaZZkKSMQyQlEoMod9xy17FmwIuEj8nZZCj1nG/gq7AaUK4wQxp3fY9acIMKUMxI+NSkGoiER6gHmlbylFCdJhN7x/DY6t0YSyULW7gVP09kaFE61ES2c4Emb6e9ybif147NfFVmFEuU0M4ni2KUwaNgJMwYJcqgg0bWYKwovZWiPtIIWxsZCUbgj//8iJpnFb8i8rZ3Xm5epvHUQSH4AicAB9cgiq4ATVQBxg8gmfwCt6cJ+fFeXc+Zq0FJ5/ZB3/gfP4AFm6WKg==</latexit>

slope⇡1

<latexit sha1_base64="zP7qZ0JqH5uABuUNi8tlx49PPa0=">AAAB/3icbVC7TsMwFHXKq5RXAImFxaJCYqoSQMBYiQW2ItGH1ITKcZ3Uqu1EtgOqQgZ+hYUBhFj5DTb+BjftAC1HutLxOffK954gYVRpx/m2SguLS8sr5dXK2vrG5pa9vdNScSoxaeKYxbITIEUYFaSpqWakk0iCeMBIOxhejv32PZGKxuJWjxLicxQJGlKMtJF69p7XoHeZeqA8h14UweKJ855ddWpOAThP3CmpgikaPfvL68c45URozJBSXddJtJ8hqSlmJK94qSIJwkMUka6hAnGi/KzYP4eHRunDMJamhIaF+nsiQ1ypEQ9MJ0d6oGa9sfif1011eOFnVCSpJgJPPgpTBnUMx2HAPpUEazYyBGFJza4QD5BEWJvIKiYEd/bkedI6rrlntZOb02r9ehpHGeyDA3AEXHAO6uAKNEATYPAInsEreLOerBfr3fqYtJas6cwu+APr8wfFGpX5</latexit>

⇧^swim ⇧^c

<latexit sha1_base64="LQGK255fM9z50tIS92jY5yGj8GU=">AAACAXicbVDLSsNAFJ34rPUVdSO4GSyCq5KoqMuCG91VsA9oYplMJ+3QeYSZiaWEuvFX3LhQxK1/4c6/cdpmoa0HLhzOuZd774kSRrXxvG9nYXFpeWW1sFZc39jc2nZ3dutapgqTGpZMqmaENGFUkJqhhpFmogjiESONqH819hsPRGkqxZ0ZJiTkqCtoTDEyVmq7+0GV3md6QPkIBop2ewYpJQfQa7slr+xNAOeJn5MSyFFtu19BR+KUE2EwQ1q3fC8xYYaUoZiRUTFINUkQ7qMuaVkqECc6zCYfjOCRVTowlsqWMHCi/p7IENd6yCPbyZHp6VlvLP7ntVITX4YZFUlqiMDTRXHKoJFwHAfsUEWwYUNLEFbU3gpxDymEjQ2taEPwZ1+eJ/WTsn9ePr09K1Vu8jgK4AAcgmPggwtQAdegCmoAg0fwDF7Bm/PkvDjvzse0dcHJZ/bAHzifP3P+luo=</latexit>

⇧^swim ! 0

<latexit sha1_base64="loqDufc2j/QAIkKqUPQUXWsJ8io=">AAAB/nicdVDLSgNBEJz1bXxFxZOXwSB4WnbN05vgRW8KRoUkhNlJxwyZnVlmesWwBPwVLx4U8ep3ePNvnMQIKlrQUFR1090VJVJYDIJ3b2p6ZnZufmExt7S8srqWX9+4sDo1HOpcS22uImZBCgV1FCjhKjHA4kjCZdQ/GvmXN2Cs0OocBwm0YnatRFdwhk5q57eaCLeYKU25NgbkWB6284XAD0pBuVqjgV8s7xfDkiOVg9pBtUpDPxijQCY4beffmh3N0xgUcsmsbYRBgq2MGRRcwjDXTC0kjPfZNTQcVSwG28rG5w/prlM6tKuNK4V0rH6fyFhs7SCOXGfMsGd/eyPxL6+RYrfWyoRKUgTFPxd1U0lR01EWtCMMcJQDRxg3wt1KeY8ZxtEllnMhfH1K/ycX+35Y8YtnpcLhySSOBbJNdsgeCUmVHJJjckrqhJOM3JNH8uTdeQ/es/fy2TrlTWY2yQ94rx+dUpac</latexit>

no correlation

<latexit sha1_base64="3mqqJe5qgficgCv3nzsrmSajCQo=">AAACAHicdVDJSgNBEO2JW4xb1IMHL41B8BQmCzHeAl70FsEskITQ06kkTXoWumvUMOTir3jxoIhXP8Obf2NnMoKKPih4vFdFVT0nkEKjbX9YqaXlldW19HpmY3Nreye7u9fUfqg4NLgvfdV2mAYpPGigQAntQAFzHQktZ3I+91s3oLTwvWucBtBz2cgTQ8EZGqmfPegi3GF0K3BMua8UyNiY9bM5O39WLVdKNrXzdow5KZbsYoUWEiVHEtT72ffuwOehCx5yybTuFOwAexFTKLiEWaYbaggYn7ARdAz1mAu6F8UPzOixUQZ06CtTHtJY/T4RMVfrqeuYTpfhWP/25uJfXifEYbUXCS8IETy+WDQMJUWfztOgA6GAo5wawrgS5lbKx0wxjiazjAnh61P6P2kW84VKvnRVztUukzjS5JAckRNSIKekRi5InTQIJzPyQJ7Is3VvPVov1uuiNWUlM/vkB6y3Twzyl2I=</latexit>

with correlation

Figure 4.7: A schematic ofΠ^{𝑎 𝑐𝑡}-Π^𝑐plot with a van-der Waals-like loop. The correlation between the collisional stress and orientation leads to∫

Π^{𝑎 𝑐𝑡}−Π_{𝑐𝑜 𝑒𝑥}^{𝑎 𝑐𝑡} 𝑑Π^𝑐 > 0 resulting in a drop in the coexistence active pressure.

bulk phases after multiplying it with𝑑Π^𝑐/𝑑 𝑧gives

∫ Π₂^𝑐

Π₁^𝑐

Π^{𝑎 𝑐𝑡}−Π_{𝑐𝑜 𝑒𝑥}^{𝑎 𝑐𝑡}

𝑑Π^𝑐 = 𝜐 2(𝑑−1)𝜁 𝐷_𝑅

∫ Π₂^𝑐

Π₁^𝑐

𝑛 Π^𝑐

𝑑 𝑑Π^𝑐

Π^𝑐 𝑛

𝑑Π^𝑐 𝑑 𝑧

2# 𝑑Π^𝑐

=− 𝜐 2(𝑑−1)𝜁 𝐷_𝑅

∫ Π^𝑐

Π^𝑐₁

𝑑 𝑑Π^𝑐

𝑛 Π^𝑐

Π^𝑐

𝑛 𝑑Π^𝑐

𝑑 𝑧 2

𝑑Π^𝑐

= 𝜐

2(𝑑−1)𝜁 𝐷_𝑅

∫ Π₂^𝑐

Π^𝑐

𝑑 𝑑Π^𝑐

Π^𝑐 𝑛

𝑑Π^𝑐 𝑑 𝑧

𝑑Π^𝑐,

(4.102) where Π^𝑐

1,2 are collisional pressures at the two points inside bulk phases and the second equation is obtained by performing the integration by parts. SinceΠ^𝑐 and Π^𝑐/𝑛 are both monotonic increasing functions of the number density, the whole integrand inside the last integral is positive. The sign of the integral indicates the predicted coexistence active pressure should be lower with the collisional-stress- orientation correlation than without it as described in fig. 4.7. It follows that collisional pressures (or corresponding number densities) of the coexisting phases are also predicted to be smaller. Because the compressibility increases with the density, the effect of change in the collisional pressure on the corresponding density is larger for large density. Therefore, the decrease in the coexisting density is more prominent in the gas-like region while it is almost negligible for the dense region.

0.1 0.2 0.3 0.4 0.5 0.6 20

40 60 80

100 120 140 160 180 200

300 320 340 360 380 400

100 120 140 160 180 200

300 320 340 360 380 400

100 120 140 160 180 200

300 320 340 360 380 400

100 120 140 160 180 200

300 320 340 360 380

<latexit sha1_base64="MJ5H9dZ22dcsIs9fPHccU6a0X7I=">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</latexit>

= 0

<latexit sha1_base64="TP+v53bSgRRsTCwTUFwHoDh3oaE=">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</latexit>

= 1

<latexit sha1_base64="T55LoFcgPWlxqK2qIWP685m/yrI=">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</latexit>

= 10

<latexit sha1_base64="kp/uUSpqrl0vuZXuc3fRwU71raM=">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</latexit>

= 10²

Figure 4.8: Binodals obtained by the mechanical theory for MIPS in 3D with the stress-orientation correlation𝜐∇ · (𝝈^𝑐𝒎/𝑛)included in the polar order flux. Here, 𝜐 is a parameter used to systematically control the weight of the correlation term in the in the flux. The correlation between the stress and orientation increases the width of the binodal as predicted. However, the widening effect saturates rapidly even at𝜐 =1.

Consequently, the predicted binodal on the activity-density plane has a broader binodal region due to the stretched gas-like branch in the presence of the collisional- stress-orientation correlation. Figure 4.8 shows that the correlation term broadens the binodal and confirms our prediction. The effect of the correlation between the stress and orientation saturates rapidly even at𝜐 ≈ 1 and does not significantly alter the binodal.

References

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[2] T. L. Hill,Statistical mechanics: principles and selected applications. Courier Corporation, 2013.

[3] M. Kardar,Statistical physics of fields. Cambridge University Press, 2007.

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Physical Review E, vol. 91, no. 3, p. 032 117, 2015.

[10] H. Row and J. F. Brady, “Reverse osmotic effect in active matter,” Physical Review E, vol. 101, no. 6, p. 062 604, 2020.

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29, 1999.

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“Active brownian particles-from individual to collective stochastic dynamics p,”The European Physical Journal Special Topics, vol. 202, 2012.

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Dalam dokumen Effect and Motility-Induced Phase Separation (Halaman 104-110)