Upgrade to Pro — share decks privately, control downloads, hide ads and more …

海洋マイクロプラスチックのサイズ分布に見られる黒体輻射との共通点

 海洋マイクロプラスチックのサイズ分布に見られる黒体輻射との共通点

波などの海洋・気象環境場が引き起こす物理的衝撃がマイクロプラスチック(通常5mm以下の破片を指す)を生成することが指摘されて久しいが、それを表現した物理モデルは提案されていない。本研究では、これら環境場が生み出す破壊エネルギーに応じてプラスチック片の粒径分布がどのように決まるかを表す理論モデルを統計力学と破壊力学に基づいて構築した。本理論モデルは次の二つの原理に基づいている:1)小さなサイズの破片形成ほど大きな破壊エネルギーを要する;2)破壊エネルギーの発生確率はボルツマン分布に従う。前者は、破壊で生じる表面エネルギーが破断面の面積に比例することからの帰結であり、後者は統計力学における有限のエネルギーの分配則から帰結である。本理論が導く粒径分布は黒体輻射のプランク分布と同形状を持ち、観測されたプラスチック片の粒径分布をマイクロプラスチック(10μm-1mmオーダー)からメソプラスチック(1mm-10cmオーダー)に渡る幅広いサイズ領域においてよく説明する。この結果は破壊一般の問題にも波及し得る。例えば、プランク分布は、冪乗則分布と対数正規分布の特徴を有しており、破壊現象に広く見られる両分布のクロスオーバーに対する一つの説明を与える可能性がある。さらに、本理論モデルは表面エネルギー密度がサイズ依存する場合にも拡張可能であり、不均一な劣化度を持つ材料の破壊現象にも適用できる可能性がある。発表時は、破壊の速度論も紹介しつつ、これら理論の発展可能性についても議論したい。

https://connpass.com/event/241136/

6734b737614a599f9b899f190e413d2d?s=128

DeepFlow, Inc.

April 29, 2022
Tweet

More Decks by DeepFlow, Inc.

Other Decks in Science

Transcript

  1. ւ༸ϚΠΫϩϓϥενοΫͷ αΠζ෼෍ʹݟΒΕΔ ࠇମ᫔ࣹͱͷڞ௨఺ ੨໦ɹ๜߂ ʢؾ৅ிʣ ڞಉݚڀऀ ݹܙ྄ɺ&UUPSF#BSCJFSJɺؙԬܟ࿨

  2. ཁࢫ ೾ͳͲͷւ༸ɾؾ৅؀ڥ৔͕Ҿ͖ى͜͢෺ཧతিܸ͕ϚΠΫϩϓϥενοΫ ௨ৗNNҎԼͷഁยΛࢦ͢ Λੜ੒͢Δ͜ ͱ͕ࢦఠ͞Εͯٱ͍͕͠ɺͦΕΛදݱͨ͠෺ཧϞσϧ͸ఏҊ͞Ε͍ͯͳ͍ɻຊݚڀͰ͸ɺ͜ΕΒ؀ڥ৔͕ੜΈग़͢ഁյ ΤωϧΪʔʹԠͯ͡ϓϥενοΫยͷཻܘ෼෍͕ͲͷΑ͏ʹܾ·Δ͔Λද͢ཧ࿦ϞσϧΛ౷ܭྗֶͱഁյྗֶʹج͍ͮ ͯߏஙͨ͠ɻຊཧ࿦Ϟσϧ͸࣍ͷೋͭͷݪཧʹج͍͍ͮͯΔɿ খ͞ͳαΠζͷഁยܗ੒΄Ͳେ͖ͳഁյΤωϧΪʔΛ ཁ͢Δʀ ഁյΤωϧΪʔͷൃੜ֬཰͸ϘϧπϚϯ෼෍ʹै͏ɻલऀ͸ɺഁյͰੜ͡Δද໘ΤωϧΪʔ͕ഁஅ໘ͷ໘ੵ

    ʹൺྫ͢Δ͜ͱ͔Βͷؼ݁Ͱ͋Γɺޙऀ͸౷ܭྗֶʹ͓͚Δ༗ݶͷΤωϧΪʔͷ෼഑ଇ͔Βؼ݁Ͱ͋Δɻຊཧ࿦͕ಋ͘ ཻܘ෼෍͸ࠇମ᫔ࣹͷϓϥϯΫ෼෍ͱಉܗঢ়Λ࣋ͪɺ؍ଌ͞ΕͨϓϥενοΫยͷཻܘ෼෍ΛϚΠΫϩϓϥενοΫ ЖNNNΦʔμʔ ͔ΒϝιϓϥενοΫ NNDNΦʔμʔ ʹ౉Δ෯޿͍αΠζྖҬʹ͓͍ͯΑ͘આ໌͢Δɻ ͜ͷ݁Ռ͸ഁյҰൠͷ໰୊ʹ΋೾ٴ͠ಘΔɻྫ͑͹ɺϓϥϯΫ෼෍͸ɺႈ৐ଇ෼෍ͱର਺ਖ਼ن෼෍ͷಛ௃Λ༗͓ͯ͠ Γɺഁյݱ৅ʹ޿͘ݟΒΕΔ྆෼෍ͷΫϩεΦʔόʔʹର͢ΔҰͭͷઆ໌Λ༩͑ΔՄೳੑ͕͋Δɻ͞Βʹɺຊཧ࿦Ϟσ ϧ͸ද໘ΤωϧΪʔີ౓͕αΠζґଘ͢Δ৔߹ʹ΋֦ுՄೳͰ͋ΓɺෆۉҰͳྼԽ౓Λ࣋ͭࡐྉͷഁյݱ৅ʹ΋ద༻Ͱ ͖ΔՄೳੑ͕͋Δɻൃද࣌͸ɺഁյͷ଎౓࿦΋঺հͭͭ͠ɺ͜ΕΒཧ࿦ͷൃలՄೳੑʹ͍ͭͯ΋ٞ࿦͍ͨ͠ɻ w
  3. ʢށాଞɼ౷ܭ෺ཧֶɼؠ೾ʣ u(λ, T)dλ = 8πhc λ5 1 ehc/λkT − 1

    dλ follows: N ¼ N0 ew A0 z ; ð1Þ where N0 denotes the concentration of microplastics collected using the neuston net, w is the plastic rise velocity (5.3 mm s−1) obtained exper- imentally by Reisser et al. (2015), and z is the vertical axis looking up- ward from the sea surface. The parameter A0 is computed as: A0 ¼ 1:5u à kHS; ð2Þ where u⁎ represents the frictional velocity of water (=0.0012 W10 ), k is the von Karman coefficient (0.4), Hs is the significant wave height, and W10 is the 10-m wind speed (Kukulka et al., 2012). Vertically integrating Eq. (1) from the sea surface (z = 0) to the infinitely deep layer (z → −∞) yields the number of microplastics per unit area M (pieces km−2) as: M ¼ N0 A0=w; ð3Þ which can be used for comparison with previous studies. We hereinaf- ter refer to M as the “total particle count” in line with Eriksen et al. (2014). To determine W10 (4.5 m s−1), the 10-m wind speeds measured by the Advanced Scatterometer (Kako et al., 2011) during the survey pe- riod were averaged spatially over the study area (lower panel of Fig. 1). In addition, to determine Hs (0.75 m), average significant wave heights during the survey period from three observatories: Wajima for waves in the Sea of Japan, Amami for the East China Sea, and Hitachi for areas east of Japan (Fig. 1), were obtained from the NOWPHAS website (http:// nowphas.mlit.go.jp/index_eng.html). In the case of the Seto Inland Sea, we used 0.34 m and 4.3 m s−1 for Hs and W10 , respectively, based on in situ data averaged over the survey period (2010–2012 summers) at the nearest observatories. To compare the estimates obtained during the present study with those of Cózar et al. (2014) in units of g km−2, the concentration n (pieces m−3) of each size (δ) was converted to weight per unit area Mw (g km−2), i.e., the “concentration” in Cózar et al. (2014) and “weight density” in Eriksen et al. (2014), which was computed as follows: Mw ¼ ρ X δ b 5 mm αδ3n   ( ) A0=w; ð4Þ where ρ denotes the density of polyethylene (950 kg m−3) and α is the “shape factor”, where 0.1 corresponds to a flat-shaped volume (Cózar et al., 2014). Note that N0 in Eq. (3) is equal to ∑ δ b 5 mm ðnÞ, and that the volume of a flat-shaped piece of microplastic with size δ (thickness of αδ) is computed as αδ3 in Eq. (4). 3. Results Generally, large marine plastic debris gradually degrades into small- three times the standard deviation from the average were removed. The only outlier eliminated by this 3σ limit was Sta. 6, where a conspic- uous quantity of microplastics (491 pieces m−3) was entangled with floating seaweed that entered the neuston net. The concentration maps (Fig. 3a and b) also demonstrate the high variability of the small plastic fragments. The concentrations of microplastics become higher both to the north of the Sea of Japan and to the south of Japan. However, negligibly small concentrations are ob- served, even at those stations neighboring those with high values. In ad- dition, of particular interest is that the concentrations reduce drastically in the southwestern part of the Sea of Japan, that is, the upstream of the Tsushima Currents (Fig. 1). The accumulation of microplastics in the downstream areas might result from unintended station placement near oceanic fronts where microplastics accumulate. However, this paper does not address the mechanism behind the accumulation of microplastics in the downstream regions of the Tsushima Currents. The distribution pattern of concentrations of mesoplastics is similar to the microplastics, although their concentrations are much smaller, irre- spective of the station. 4. Discussion 4.1. Are the East Asian seas recognized as a hot spot of microplastics? Fig. 2. Size distribution of small plastic fragments. The bars indicate the concentration in each size range on the abscissa. Note that the intervals of size ranges are 0.1 mm for microplastics, 1 mm for mesoplastics b10 mm, and 10 mm for mesoplastics N10 mm. 620 A. Isobe et al. / Marine Pollution Bulletin 101 (2015) 618–623 1MBODL`TGPSNVMB 0VSGPSNVMB S(λ)dλ = A λ4 1 eb/γ − 1 dλ ʢ*TPCFFUBMʣ
  4. REPORTS AND STUDIES 99 GUIDELINES FOR THE MONITORING AND ASSESSMENT

    OF PLASTIC LITTER IN THE OCEAN IMO FAO UNESCO IOC WMO UNIDO IAEA UN UN ENVIRONMENT ISA UNDP Notes: GESAMP is an advisory body consisting of specialized experts nominated by the Sponsoring Agencies -13*%392)7'3-3'92-(3;13-%)%9292)492(4 -XWTVMRGMTEPXEWOMWXSTVSZMHIWGMIRXMƤGEHZMGIGSRGIVRMRKXLI TVIZIRXMSRVIHYGXMSRERHGSRXVSPSJXLIHIKVEHEXMSRSJXLIQEVMRIIRZMVSRQIRXXSXLIɄ7TSRWSVMRK%KIRGMIW The report contains views expressed or endorsed by members of GESAMP who act in their individual capacities; their views may not necessarily correspond with those of the Sponsoring Agencies. Permission may be granted by any of the Sponsoring Agencies for the report to be wholly or partially reproduced in publication by any individual who is not a staff member of a Sponsoring Agency of GESAMP, provided that the source of the extract and the condition mentioned above are indicated. Information about GESAMP and its reports and studies can be found at: http://gesamp.org ISSN 1020-4873 (GESAMP Reports & Studies Series) Copyright © IMO, FAO, UNESCO-IOC, UNIDO, WMO, IAEA, UN, UNEP, UNDP, ISA 2019 For bibliographic purposes this document should be cited as: GESAMP (2019). Guidelines or the monitoring and assessment of plastic litter and microplastics in the ocean (Kershaw P.J., Turra A. and Galgani F. editors), (IMO/FAO/UNESCO-IOC/UNIDO/WMO/IAEA/UN/UNEP/UNDP/ISA Joint Group of Experts on XLI7GMIRXMƤG%WTIGXWSJ1EVMRI)RZMVSRQIRXEP4VSXIGXMSR 6IT7XYH+)7%142ST Editors Peter Kershawa, Alexander Turrab and Francois Galganic Working Group members Akbar Tahir, Alexander Turra, Amy Lusher, Chris Wilcox, Denise Hardesty, Francois Galgani, Hideshige Takada, Marcus Erikson, Martin Hassellov, Martin Thiel, Peter Kershaw, Sang Hee Hong, Sheri (Sam) Mason, Weiwei Zhang and Won Joon Shim Observers %Q]9LVMR 23%% ,IRVMO)RIZSPHWIR -3' .SERE%OVSƤ 92)4 ERH,IMHM7EZIPPM7SRHIVFIVK 92)4 Disclaimer: The guidance and recommendations provided in this report are intended for use by competent bodies and their employees operating within the customs, norms and laws of their respective countries. Collectively, GESAMP, the report editors and report contributors do not accept any liability resulting from the use of these guidelines. Users are encouraged to follow appropriate health and safety provisions and adopt safe working practices for working in and around the marine environment and in follow-up sample processing and analysis, especially, but not limited to: sampling from vessels at sea, diving operations, Published by United Nations Environment Programme (UNEP) Printed by 9RMXIH2EXMSRW3JƤGI2EMVSFM 9232 4YFPMWLMRK7IVZMGIW7IGXMSR?)17-73'IVXMƤIHA Layout: Eugene Papa - UNON, Publishing Services Section Cover photo: © O.DUGORNAY/IFREMER ISSN: 1020-4873 Copyright © IMO, FAO, UNESCO-IOC, UNIDO, WMO, IAEA, UN, UN Environment, UNDP, ISA 2019 Since the 1950s, when large-scale production of plastics began, an incresing proportion of solid waste in the ocean has consisted of this material, representeing up to 80% of marine litter found in surveys (UNEP, 2016) — ๯಄ΑΓҾ༻
  5. ւ༸ϓϥενοΫ໰୊௒֓؍ 'JTIJOHOFU 1MBTUJDCBH 1MBTUJDCPUUMF  1MBTUJDQSPEVDUT )VNBOCPEZ &NJTTJPOPGQMBTUJDHBSCBHF 4501FNJTTJPO TPDJBMBDUJWJUZ

    ւͷ໰୊ ཮ͷ໰୊ .JDSPQMBTUJDT .JDSPCFBET 'JCFS  Ұ࣍ϓϥενοΫ ೋ࣍ϓϥενοΫ 'SBHNFOUT 5BOBLBBOE5BLBEB 4DJ3FQ ࣸਅ͸
  6. SCIENCE sciencemag.org 13 FEBRUARY 2015 • VOL 347 ISSUE 6223

    769 calculated for populations within 50 km of the coast in the 192 countries considered. pop., population; gen., generation; ppd, person per day; MMT, million metric tons. Rank Country Econ. classif. Coastal pop. [millions] Waste gen. rate [kg/ppd] % plastic waste % mismanaged waste Mismanaged plastic waste [MMT/year] % of total mismanaged plastic waste Plastic marine debris [MMT/year] 1 China UMI 262.9 1.10 11 76 8.82 27.7 1.32–3.53 2 Indonesia LMI 187.2 0.52 11 83 3.22 10.1 0.48–1.29 3 Philippines LMI 83.4 0.5 15 83 1.88 5.9 0.28–0.75 4 Vietnam LMI 55.9 0.79 13 88 1.83 5.8 0.28–0.73 5 Sri Lanka LMI 14.6 5.1 7 84 1.59 5.0 0.24–0.64 6 Thailand UMI 26.0 1.2 12 75 1.03 3.2 0.15–0.41 7 Egypt LMI 21.8 1.37 13 69 0.97 3.0 0.15–0.39 8 Malaysia UMI 22.9 1.52 13 57 0.94 2.9 0.14–0.37 9 Nigeria LMI 27.5 0.79 13 83 0.85 2.7 0.13–0.34 10 Bangladesh LI 70.9 0.43 8 89 0.79 2.5 0.12–0.31 11 South Africa UMI 12.9 2.0 12 56 0.63 2.0 0.09–0.25 12 India LMI 187.5 0.34 3 87 0.60 1.9 0.09–0.24 13 Algeria UMI 16.6 1.2 12 60 0.52 1.6 0.08–0.21 14 Turkey UMI 34.0 1.77 12 18 0.49 1.5 0.07–0.19 15 Pakistan LMI 14.6 0.79 13 88 0.48 1.5 0.07–0.19 16 Brazil UMI 74.7 1.03 16 11 0.47 1.5 0.07–0.19 17 Burma LI 19.0 0.44 17 89 0.46 1.4 0.07–0.18 18* Morocco LMI 17.3 1.46 5 68 0.31 1.0 0.05–0.12 19 North Korea LI 17.3 0.6 9 90 0.30 1.0 0.05–0.12 20 United States HIC 112.9 2.58 13 2 0.28 0.9 0.04–0.11 *If considered collectively, coastal European Union countries (23 total) would rank eighteenth on the list on August 20, 2019 SCIENCE sciencemag.org calculated for populations within 50 km of the coast in the metric tons. Rank Country Econ. classif. Coastal pop. [millions] Was [kg 1 China UMI 262.9 2 Indonesia LMI 187.2 3 Philippines LMI 83.4 4 Vietnam LMI 55.9 5 Sri Lanka LMI 14.6 6 Thailand UMI 26.0 7 Egypt LMI 21.8 8 Malaysia UMI 22.9 9 Nigeria LMI 27.5 10 Bangladesh LI 70.9 11 South Africa UMI 12.9 12 India LMI 187.5 0 13 Algeria UMI 16.6 14 Turkey UMI 34.0 15 Pakistan LMI 14.6 16 Brazil UMI 74.7 17 Burma LI 19.0 18* Morocco LMI 17.3 19 North Korea LI 17.3 20 United States HIC 112.9 *If considered collectively, coastal European Union countries (Jambeck et al 2015)           
  7. ྼԽͱഁյ ഁյ ඿Ͱਐߦ likely to initially biodegrade rapidly. In any

    event the finding is of little practical consequence. Embrittlement in beach weathering increases the specific surface area of the plastics by several orders of magnitude and this might be expected to increase its rate of bio- degradation (Kawai et al., 2004). But, this small increase in the rate of an already very slow process to effect its complete mineralisa- tion in a reasonable timescale of a few years. The laboratory results are generally consistent with the findings from field exposures; HDPE, LDPE and PP coupons immersed in Bay of Bengal (India) observed over a 6-month periods in a recent study. Maximum w an 19 su te M Su ti Ta an Fig. 4. Right: PP exposed to a 600 watt xenon source for 6 weeks (Yakimets et al. 2004); Mid 1993); Left: LDPE weathered in a weatherometer for 800h (Küpper, et al., 2004). ௿ີ౓ϙϦΤνϨϯ IXFBUIFSPNFUFS ,VQQFSFUBM lࢵ֎ઢ౳ʹΑͬͯྼԽ͠ɺ ೾ͳͲͷ෺ཧతিܸʹΑͬͯഁյ͞ΕΔz ŠŠϚΠΫϩϓϥενοΫͷੜ੒ʹ͍ͭͯͷҰൠతݟղ Figure 4: Schematic view of system in terms of statistical mechanics. crush energy: p(ε) ∝ e−ε/γ, (2) where γ may be regarded as a representative value of the energy of the natural phenom- ena. Eq. 2 indicates that a crush event with a large energy are less frequent, consistent with our usual experience. In the statiscital mechanics, the Boltzmann distribution ࢵ֎ઢ ྼԽ ഁյ 67SBEJBUJPO ೾ 8BWFT %FHSBEBUJPO 'SBDUVSF ྼԽ ֎తཁҼ
  8. ༷ʑͳܗɾछྨɾ৔ॴ ҰճͷαϯϓϦϯάͰಘΒΕ༷ͨʑͳछྨͷϓϥενοΫ ԙ͔͍Ί͍ߤւ ௿ີ౓ϙϦΤνϨϯ ߴີ౓ϙϦΤνϨϯ ϙϦϓϩϐϨϯ ϙϦԘԽϏχϧ ϙϦενϨϯ ̑େ൚༻ϓϥ )BSE4PGU

    0QFORVFTUJPOT 'SBHNFOUBUJPOQSPDFTT "EIFTJP Journal of Geophysical Re FEBRUARY 2004 T S U J I N O A N TABLE 1. Subduction rate for the mode w Density (s u ) Detrainment (Sv yr21) Reentrainment (Sv yr21) Subductio (Sv yr21) 25.05 25.15 25.25 25.35 25.45 3.8 5.1 6.4 5.9 4.6 0.7 1.1 1.2 1.1 1.1 3.1 4.0 5.2 4.8 3.5 25.55 25.65 25.75 25.85 25.95 3.9 3.7 4.3 4.7 6.2 1.1 1.5 2.1 2.7 2.8 2.8 2.2 2.2 2.0 3.4 26.05 26.15 26.25 26.35 26.45 5.6 4.9 4.5 6.0 7.0 2.9 2.8 2.8 4.0 2.7 2.7 2.1 1.7 2.0 4.3 (1996). Its northern boundary is the outcrop of the sub- arctic permanent halocline that prohibits deepening of Environmental Science & Technology Impacts of Biofilm Formation on the Fate and Potential Effects of Microplastic in the Aquatic Environment Christoph D. Rummel,*,† Annika Jahnke,‡ Elena Gorokhova,§ Dana Kühnel,† and Mechthild Schmitt-Jansen*,† † Department of Bioanalytical Ecotoxicology, Helmholtz Centre for Environmental Research-UFZ, Permoserstrasse 15, 04318 Leipzig, Germany ‡ Department of Cell Toxicology, Helmholtz Centre for Environmental Research-UFZ, Permoserstrasse 15, 04318 Leipzig, Germany §Department of Environmental Science & Analytical Chemistry (ACES), Stockholm University, Svante Arrhenius väg 8, 106 91 Stockholm, Sweden * S Supporting Information ABSTRACT: In the aquatic environment, microplastic (MP; <5 mm) is a cause of concern because of its persistence and potential adverse effects on biota. Studies of microlitter impacts are mostly based on virgin and spherical polymer particles as model MP. However, in pelagic and benthic environments, surfaces are always colonized by microorgan- isms forming so-called biofilms. The influence of such biofilms on the fate and potential effects of MP is not understood well. Here, we review the physical interactions of early microbial colonization on plastic surfaces and their reciprocal influence on the weathering processes and vertical transport as well as sorption and release of contaminants by MP. Possible ecological consequences of biofilm formation on MP, such as trophic transfer of MP particles and potential adverse effects of MP, are virtually unknown. However, evidence is accumulating that the biofilm−plastic interactions have the capacity to influence the fate and impacts of MP by modifying the physical properties of the particles. There is an urgent research need to better understand these interactions and increase the ecological relevance of current laboratory testing by simulating field conditions in which microbial life is a key driver of biogeochemical processes. ▪ INTRODUCTION In the aquatic environment, plastic litter has emerged as a major pollution issue, because it is only slowly degradable,1,2 is have so far reported any ecologically plausible adverse effects of MP on primary consumers, we know very little about the interactions between these particles and their potential Review pubs.acs.org/journal/estlcu This is an open access article published under an ACS AuthorChoice License, which permits copying and redistribution of the article or any adaptations for non-commercial purposes. Downloaded via 106.167.208.166 on May 18, 2021 at 05:19:16 (UTC). ttps://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.   B )JTUPHSBN C 4QFDUSBMEFOTJUZ 4USPOH8FBL ܗ छ ྨ ৔ ॴ
  9. ϚΠΫϩϓϥͷαΠζ෼෍ <ࡱӨɿ੨໦ɾ౻ݩ ԙ͔͍Ί͍ߤւ > ϚΠΫϩϓϥενοΫ࠾ूͷ༷ࢠ ð1Þ ected using the btained

    exper- xis looking up- d as: ð2Þ 0012 W10 ), k is ave height, and ally integrating layer (z → −∞) es km−2) as: ð3Þ s. We hereinaf- h Eriksen et al. eeds measured the survey pe- panel of Fig. 1). t wave heights ma for waves in hi for areas east ebsite (http:// he Seto Inland ectively, based 012 summers) ent study with oncentration n three times the standard deviation from the average were removed. The only outlier eliminated by this 3σ limit was Sta. 6, where a conspic- uous quantity of microplastics (491 pieces m−3) was entangled with floating seaweed that entered the neuston net. The concentration maps (Fig. 3a and b) also demonstrate the high variability of the small plastic fragments. The concentrations of microplastics become higher both to the north of the Sea of Japan and Fig. 2. Size distribution of small plastic fragments. The bars indicate the concentration in each size range on the abscissa. Note that the intervals of size ranges are 0.1 mm for microplastics, 1 mm for mesoplastics b10 mm, and 10 mm for mesoplastics N10 mm. et al. / Marine Pollution Bulletin 101 (2015) 618–623 *TPCFFUBM a. b. c. d.
  10. ϚΠΫϩϓϥͷαΠζ෼෍ ႈ৐ଇ <$P[ÂSFUBM> ð1Þ ected using the btained exper- xis

    looking up- d as: ð2Þ 0012 W10 ), k is ave height, and ally integrating layer (z → −∞) es km−2) as: ð3Þ s. We hereinaf- h Eriksen et al. eeds measured the survey pe- panel of Fig. 1). t wave heights ma for waves in hi for areas east ebsite (http:// he Seto Inland ectively, based 012 summers) ent study with oncentration n three times the standard deviation from the average were removed. The only outlier eliminated by this 3σ limit was Sta. 6, where a conspic- uous quantity of microplastics (491 pieces m−3) was entangled with floating seaweed that entered the neuston net. The concentration maps (Fig. 3a and b) also demonstrate the high variability of the small plastic fragments. The concentrations of microplastics become higher both to the north of the Sea of Japan and Fig. 2. Size distribution of small plastic fragments. The bars indicate the concentration in each size range on the abscissa. Note that the intervals of size ranges are 0.1 mm for microplastics, 1 mm for mesoplastics b10 mm, and 10 mm for mesoplastics N10 mm. et al. / Marine Pollution Bulletin 101 (2015) 618–623 *TPCFFUBM a. b. c. d. ႈ৐ଈ αΠζͷେ͖͍ྖҬͰΑ͍Ұக ʢϝιϓϥενοΫʣ
  11. ϚΠΫϩϓϥͷαΠζ෼෍ ð1Þ ected using the btained exper- xis looking up-

    d as: ð2Þ 0012 W10 ), k is ave height, and ally integrating layer (z → −∞) es km−2) as: ð3Þ s. We hereinaf- h Eriksen et al. eeds measured the survey pe- panel of Fig. 1). t wave heights ma for waves in hi for areas east ebsite (http:// he Seto Inland ectively, based 012 summers) ent study with oncentration n three times the standard deviation from the average were removed. The only outlier eliminated by this 3σ limit was Sta. 6, where a conspic- uous quantity of microplastics (491 pieces m−3) was entangled with floating seaweed that entered the neuston net. The concentration maps (Fig. 3a and b) also demonstrate the high variability of the small plastic fragments. The concentrations of microplastics become higher both to the north of the Sea of Japan and Fig. 2. Size distribution of small plastic fragments. The bars indicate the concentration in each size range on the abscissa. Note that the intervals of size ranges are 0.1 mm for microplastics, 1 mm for mesoplastics b10 mm, and 10 mm for mesoplastics N10 mm. et al. / Marine Pollution Bulletin 101 (2015) 618–623 *TPCFFUBM a. b. c. d. ႈ৐ଈ ର਺ਖ਼ن෼෍ αΠζͷେ͖͍ྖҬͰΑ͍Ұக αΠζͷখ͍͞ྖҬͰΑ͍Ұக ʢϝιϓϥενοΫʣ ΫϩεΦʔόʔͷՄೳੑ
  12. ϞσϦϯάͰͷཹҙ఺ w֎తཁҼΛύϥϝʔλԽͰ͖Δ wܗɾछྨɾ৔ॴʹΑΒͳ͍ wΫϩεΦʔόʔʢႈ৐ଈͱର਺ਖ਼ن෼෍ʣΛ આ໌Ͱ͖Δ

  13. ΤωϧΪʔͱ͍͏ࢹ఺ ෼཭ Ҿྗͱੱྗ͕௼Γ߹͏ڑ཭ ݁߹ΤωϧΪʔ ද໘ΤωϧΪʔ ʼ ʙ ഁյʹඞཁͳΤωϧΪʔ

  14. Ϟσϧ

  15. ᶃখ͞ͳഁยͷܗ੒΄Ͳେ͖ͳഁյΤωϧΪʔΛཁ͢Δ ᶄഁյΤωϧΪʔͷൃੜ֬཰͸ϘϧπϚϯ෼෍ʹै͏ ೋͭͷݪཧ

  16. A plate with the size of L2. L L Crush

    energy ε 破壊 エネルギー
  17. A plate with the size of L2. L L λ

    λ fragments into n x n cells ν = n L 8BWFOVNCFSɿ λ = L n 8BWFMFOHUIɿ Cell size ɾɾɾ QJFDFT O QJFDFT O QJFDFT O ͨͱ͑͹ʜ Crush energy /VNCFSPGGSBHNFOUT GPSFBDITJ[F (n × n) ⋅ j ʜαΠζ෼෍ ε 破壊 エネルギー j = 1 j = 2 j = 3 ɾɾɾ Crush energy Number of fractured plates (j = 1,2,3,⋯) ͋Δn ʹ஫໨ͨ͠ͱ͖ɺ ෼ׂn YnͳΔΠϕϯτ ͸Ұճͱ͸ݶΒͳ͍ 破壊 エネルギー
  18. W ϕ A A Work done by mechanical force Surface

    energy ഁյΤωϧΪʔͱ͸ #SFBLBHF ɾɾɾ l = 2L l = 4L l = 6L O O O l = 2(n − 1)L = 4Lν − 2L “Surface” is the total length of the contact boundaries Total length Proportional to ν ϕ A ⟶ bν 4VSGBDFFOFSHZGPSPOFQMBUF <GPSBDPOTUBOUFOFSHZEFOTJUZ> GPSjQMBUFT ε ≡ jbν (b = const.) ϕ A ≤ W ε ʜഁյΤωϧΪʔʢධՁ஋ʣ
  19. ϘϧπϚϯ෼෍ഁյΤωϧΪʔͷੜى֬཰ p(ε) ∝ e−ε/γ, (2) a representative value of the

    energy of the natural phenom- crush event with a large energy are less frequent, consistent In the statiscital mechanics, the Boltzmann distribution t the energy of a subsystem surrounded by a heat bath fol- of the energies between them. Here, the plastics and the rded as analogous to the subsystem and heat bath, respec- obability takes discrete values due to the discretized form n (1) (Fig. 5). Using (1) and (2), the expected value of the Ѝ Џ Q Џ 環境場のエネルギー S = − k ∫ ∞ 0 [p(E)ln p(E)]Ω(E)dE ⟨E⟩ = ∫ ∞ 0 Ep(E)Ω(E)dE ∫ ∞ 0 p(E)dE = 1 p(E) = e−βE Z , Z = ∫ ∞ 0 e−βEΩ(E)dE Τϯτϩϐʔɿ ΤωϧΪʔɿ ن֨Խ৚݅ɿ δp : [β = 1/kT] Š೤ྗֶؔ܎ࣜΑΓ
  20. ϘϧπϚϯ෼෍ഁյΤωϧΪʔͷੜى֬཰ ⟨ε⟩ ν = ∑∞ j=1 jbνe−jbν ∑∞ j=1 e−jbν

    #PTFEJTUSJCVUJPO ⟨j⟩ ν = ⟨ε⟩ ν bν = 1 ebν/γ − 1 ഁյΤωϧΪʔͷظ଴஋ յΕΔ൘ͷ਺ͷظ଴஋ ε = jbν ЗݻఆͷԼͰ p(ε) ∝ e−ε/γ, (2) a representative value of the energy of the natural phenom- crush event with a large energy are less frequent, consistent In the statiscital mechanics, the Boltzmann distribution t the energy of a subsystem surrounded by a heat bath fol- of the energies between them. Here, the plastics and the rded as analogous to the subsystem and heat bath, respec- obability takes discrete values due to the discretized form n (1) (Fig. 5). Using (1) and (2), the expected value of the Ѝ Џ Q Џ 環境場のエネルギー
  21. [A : arbitrary constant] Figure 6: Size distributions expected from

    (5) for different γ with intervals of 0.1 under fixed A (A = 1.0). The size distribution with a large magnitude corresponds to a large The maxima of the respective size distributions (shown in dot) follows the dashed curve. (n × n) ⋅ j /VNCFSPGGSBHNFOUTGPSFBDITJ[F and we end up with Eν [ε] = bνEν [j], where Eν [j] = 1 ebν/γ − 1 (3) is the expected value of the number of fractured plates for each ν. This formula is the so-called Bose distribution [7]. Recall that the number of the fragments of a particular size is given by the product of the quantities n×n and the number of fractured plates. The former quantity can be written as 4L2ν2 by the definition of the wavenumber and the latter can be given by the expected value of the fractured plates (Eq. 3). Accordingly, the number of fragments 7 ⟨j⟩ P(ν)dν = Aν2 1 ebν/γ − 1 dν S(λ)dλ = A λ4 1 eb/γλ − 1 dλ αΠζεϖΫτϧ For ν : For λ : ʢށాଞɼ౷ܭ෺ཧֶɼؠ೾ʣ λ MBSHFЍ TNBMMЍ
  22. b. d. a. b. c. d. ؍ଌσʔλͱͷൺֱ <ൺֱσʔλɿ*TPCFFUBM> ೔ຊपล w

    ؍ଌ஋ͱΑ͘߹͏
  23. b. d. a. b. c. d. ؍ଌσʔλͱͷൺֱ <ൺֱσʔλɿ*TPCFFUBM> ೔ຊपล w

    ؍ଌ஋ͱΑ͘߹͏ w ؍ଌσʔλʹΑΒͳ͍ w શͯͷαΠζྖҬΛΧόʔ a. b. c. d. a. b. c. d. Figure 3: a) Basin-wise size spectral density of microplastics abundan (their Fig.S6) and that expected from our model. b) Sum of the size sp for observation (black dots) and our model (blue curve). Orange curv lognormal distribution and a cube power law, respectively. Gray dots a dots and blue curve, respectively, except that the South Atlantic data conducted for the sizes less than 30 mm because of limitation of dig from the figure in Cózar et al. of the plastic fragments in their study is similar to Isobe et al [6 model generally reproduces well the size distribution of each ba derestimates in the sizes less than 0.5 mm in the South Pacific O range in the South Atlantic and North Pacific Oceans (Fig. 3b). I duces that the size distribution comes to follow the cube power range (Blue curve in Fig. 3d). Non-negligible differences betw <ൺֱσʔλɿ$Ó[BSFUBM> ੈքͷΰϛύον
  24. ؍ଌσʔλͱͷൺֱ a. b. c. d. a. b. c. d. Figure

    3: a) Basin-wise size spectral density of microplastics abundan (their Fig.S6) and that expected from our model. b) Sum of the size sp for observation (black dots) and our model (blue curve). Orange curv lognormal distribution and a cube power law, respectively. Gray dots a dots and blue curve, respectively, except that the South Atlantic data conducted for the sizes less than 30 mm because of limitation of dig from the figure in Cózar et al. of the plastic fragments in their study is similar to Isobe et al [6 model generally reproduces well the size distribution of each ba derestimates in the sizes less than 0.5 mm in the South Pacific O range in the South Atlantic and North Pacific Oceans (Fig. 3b). I duces that the size distribution comes to follow the cube power range (Blue curve in Fig. 3d). Non-negligible differences betw <ൺֱσʔλɿ$Ó[BSFUBM> ੈքͷΰϛύον w ؍ଌ஋ͱΑ͘߹͏ w ؍ଌσʔλʹΑΒͳ͍ w શͯͷαΠζྖҬΛΧόʔ <ൺֱσʔλɿ1BCPSUTBWBBOE-BNQJUU> B C D E େ੢༸ͷѥද૚
  25. ࠇମ᫔ࣹͱͷܗࣜతͳൺֱ ε = jbν ඈͼඈͼͷΤωϧΪʔ ϵ = nℏω ࠇମ᫔ࣹ ຊϞσϧ

    ϘϧπϚϯ෼෍ ෼෍ࣜ ೾ͷ਺ີ౓ º Ϙʔζ෼෍  º ϑΥτϯͷΤωϧΪʔ ഁย਺ º Ϙʔζ෼෍ ω2 π2c3 ℏω eℏω/kT − 1 Aν2 1 ebν/γ − 1 ʢௐ࿨ৼಈࢠͷྔࢠԽʣ ʢյΕΔ൘ͷ਺ʣ p(ϵ) ∝ e−ϵ/kT p(ε) ∝ e−ε/γ
  26. ݽཱܥ Թ౓ T

  27. Figure 4: Schematic view of system in terms of statistical

    mechanics. crush energy: p(ε) ∝ e−ε/γ, (2) where γ may be regarded as a representative value of the energy of the natural phenom- Figure 4: Schematic view of system in terms of statistical mechanics. Figure 4: Schematic view of system in terms of statistical mechanics. crush energy: p(ε) ∝ e−ε/γ, (2) where γ may be regarded as a representative value of the energy of the natural phenom- ena. Eq. 2 indicates that a crush event with a large energy are less frequent, consistent with our usual experience. In the statiscital mechanics, the Boltzmann distribution provides the probability that the energy of a subsystem surrounded by a heat bath fol- lows under the conservation of the energies between them. Here, the plastics and the natural phenomena are regarded as analogous to the subsystem and heat bath, respec- Figure 4: Schematic view of system in terms of statistical mechanics. crush energy: Figure 4: Schematic view of system in terms of statistical mechanics. crush energy: Figure 4: Schematic view of system in terms of statistical mechanics. crush energy: p(ε) ∝ e−ε/γ, (2) where γ may be regarded as a representative value of the energy of the natural phenom- Figure 4: Schematic view of system in terms of statistical mechanics. Figure 4: Schematic view of system in terms of statistical mechanics. crush energy: p(ε) ∝ e−ε/γ, (2) where γ may be regarded as a representative value of the energy of the natural phenom- ena. Eq. 2 indicates that a crush event with a large energy are less frequent, consistent with our usual experience. In the statiscital mechanics, the Boltzmann distribution provides the probability that the energy of a subsystem surrounded by a heat bath fol- lows under the conservation of the energies between them. Here, the plastics and the natural phenomena are regarded as analogous to the subsystem and heat bath, respec- tively (Fig. 4). Now, the probability takes discrete values due to the discretized form Figure 4: Schematic view of system in terms of statistical mechanics. crush energy: p(ε) ∝ e−ε/γ, (2) where γ may be regarded as a representative value of the energy of the natural phenom- ena. Eq. 2 indicates that a crush event with a large energy are less frequent, consistent with our usual experience. In the statiscital mechanics, the Boltzmann distribution provides the probability that the energy of a subsystem surrounded by a heat bath fol- lows under the conservation of the energies between them. Here, the plastics and the natural phenomena are regarded as analogous to the subsystem and heat bath, respec- tively (Fig. 4). Now, the probability takes discrete values due to the discretized form of the crush energy defined in (1) (Fig. 5). Using (1) and (2), the expected value of the ∞ ∞ Figure 4: Schematic view of system in terms of statistical mechanics. crush energy: p(ε) ∝ e−ε/γ, (2) where γ may be regarded as a representative value of the energy of the natural phenom- ena. Eq. 2 indicates that a crush event with a large energy are less frequent, consistent Figure 4: Schematic view of system in terms of statistical mec ؀ڥ৔ͷΤωϧΪʔ ʢ೾ͳͲʣ γ ݽཱܥ
  28. ల๬

  29. ද໘ΤωϧΪʔີ౓͸Ұఆʁ ε = jbν ϕ A A Surface energy ϕ

    A → ϕΔh l b ν ഁյΤωϧΪʔ Δh ε = j ∫ A ϕ(x; ν)dx ຊདྷ͸͜͏ॻ͘΂͖ʁʁ ε = jbν ϕ A A Surface energy ϕ A → ϕΔh l b ν ഁյΤωϧΪʔ Δh ɾ  ɾ  ɾ   ɾ  ɾ       ෆۉҰͳྼԽ αΠζґଘ͢ΔྼԽ      ϕ(x) ϕ(ν)    ද໘ΤωϧΪʔີ౓͸Ұఆʁ ε = jbν ϕ A A Surface energy ϕ A → ϕΔh l b ν ഁյΤωϧΪʔ Δh ɾ  ɾ  ɾ   ɾ  ɾ  [x : positional vector]
  30. ଎౓࿦ ܦ࣌มԽ ͱͷରԠ ;J⒎.D(SBEZ ∂F(r, t) ∂t = − F(r,

    t) ∫ r 0 W(r → r′ )dr′ + ∫ ∞ r F(r′ , t)W(r′ → r)dr′ J. Phys. D: Appl. Phys. 41 (2008) 085405 M 0 0.2 0.4 0.6 0.8 1 1.2 size, r 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 f(r) f(r) Modelling of the evolution equation α distribution - Gaussian (0.7; 0.07) initial radius distribution - Gaussian (1; 0.1) frequency = 100 Hz non-dimensional times: 0; 1.1; 3.3; 4.4; 6.6; 7.7 0 0.2 0.4 0.6 0.8 1 size,r Solution of the Fokker–Planck eq <ln(α)>=–0.36; <ln**2(α)>=0.14 initial radius distribution - gaussia frequency=100 Hz non-dimensional times: 0; 1.1; 3.3 (Gorokhovski & Seveliev 2008) q(α)   ֬ ཰ ີ ౓ αΠζ ݮগ཰ c 0 ∂F(r, t) ∂t = − ν(r)F(r, t) + c 0 ∫ 1 0 ν ( r α) F ( r α) q(α) dα α ස౓ ݮগ཰ ഁย਺ ( ln α νt) z2k+1 ( ln2 α νt)2k+1 = t→∞ k(2k + 1)!! ln2 α νt ln3 α 3 ln2 α 1 + O 1 t , (12) where O(1/t) denotes smaller terms of order 1/t, and lnl α = 1 0 lnl αq(α) dα. In these expressions, we use ( )!! to denote the semi-factorial function (2k − 1)!! = 1 · 3 · 5 · . . . · 2k − 1, (2k + 1)!! = 1 · 3 · 5 · . . . · (2k + 1), it being understood that (−1)!! = 1! = 1. Comparing of moments (11), (12) with even and odd moments of the Gaussian distribution G(z) = (1/ √ 2πσ)e−z2/2σ2 , which are z2k /σ2k = (2k − 1)!! and z2k+1 /σ2k+1 = 0, shows that for large times, the central logarithmic moments of the solution of equation (6) tend to moments of the Gaussian distribution. As a result, the asymptotic solution of equation (6) can be derived: f (r, t) = t→∞ 1 R 1 2π ln2 α νt exp − ln α 2 2 ln2 α νt × exp − (ln(r/R))2 2 ln2 α νt R r 1−( ln α / ln2 α ), (13) where R is the initial length scale, R = e ln r t0 . This expression confirms the main result of Kolmogorov [1] 2 ln α r As time tends to infinity, this distribution tends to the pow distribution: f (r, t) ∼ t→∞ 1 r 1−( ln α / ln2 α ) . (1 It should be noted that the power distribution (17) is only va for r > rcr(t), where rcr(t) is a lower boundary of leng scales, which decreases exponentially with time: rcr(t) → when t → ∞. So that the power distribution of the form f (r, t|rcr) = βrβ crr−1−β, r > rcr, 0, r < rcr, with β > 0, tends to the Dirac delta function as time goes infinity: f (r, t|rcr) → rcr→0 δ(r). This result follows from the fact that βrβ cr ∞ rcr r−1−β dr = 1 Let us make another important remark. Previous stud (e.g. [10,11]) stated that when the dispersion (ln r − ln r ) is large, the ratio ln r / (ln r − ln r )2 is small an therefore, the log-normal distribution could be mimicked by 1/r distribution. However, one can show that first logarithm moments of the solution of equation (6) evolve as <f ͸F ͷن֨Խ> (Gorokhovski & Seveliev 2008) (PSPLIPWTLJ4FWFMJFW  ͷղ<ස౓Ұఆ> ཻܘ෼෍ͷ࣌ؒมԽ
  31. ଎౓࿦ ܦ࣌มԽ ͱͷରԠ ( ln α νt) z2k+1 ( ln2

    α νt)2k+1 = t→∞ k(2k + 1)!! ln2 α νt ln3 α 3 ln2 α 1 + O 1 t , (12) where O(1/t) denotes smaller terms of order 1/t, and lnl α = 1 0 lnl αq(α) dα. In these expressions, we use ( )!! to denote the semi-factorial function (2k − 1)!! = 1 · 3 · 5 · . . . · 2k − 1, (2k + 1)!! = 1 · 3 · 5 · . . . · (2k + 1), it being understood that (−1)!! = 1! = 1. Comparing of moments (11), (12) with even and odd moments of the Gaussian distribution G(z) = (1/ √ 2πσ)e−z2/2σ2 , which are z2k /σ2k = (2k − 1)!! and z2k+1 /σ2k+1 = 0, shows that for large times, the central logarithmic moments of the solution of equation (6) tend to moments of the Gaussian distribution. As a result, the asymptotic solution of equation (6) can be derived: f (r, t) = t→∞ 1 R 1 2π ln2 α νt exp − ln α 2 2 ln2 α νt × exp − (ln(r/R))2 2 ln2 α νt R r 1−( ln α / ln2 α ), (13) where R is the initial length scale, R = e ln r t0 . This expression confirms the main result of Kolmogorov [1] 2 ln α r As time tends to infinity, this distribution tends to the pow distribution: f (r, t) ∼ t→∞ 1 r 1−( ln α / ln2 α ) . (1 It should be noted that the power distribution (17) is only va for r > rcr(t), where rcr(t) is a lower boundary of leng scales, which decreases exponentially with time: rcr(t) → when t → ∞. So that the power distribution of the form f (r, t|rcr) = βrβ crr−1−β, r > rcr, 0, r < rcr, with β > 0, tends to the Dirac delta function as time goes infinity: f (r, t|rcr) → rcr→0 δ(r). This result follows from the fact that βrβ cr ∞ rcr r−1−β dr = 1 Let us make another important remark. Previous stud (e.g. [10,11]) stated that when the dispersion (ln r − ln r ) is large, the ratio ln r / (ln r − ln r )2 is small an therefore, the log-normal distribution could be mimicked by 1/r distribution. However, one can show that first logarithm moments of the solution of equation (6) evolve as <f ͸F ͷن֨Խ> (Gorokhovski & Seveliev 2008) S N (r) = b3 σγ3 1 r4 1 eb/rγ − 1 "PLJ'VSVF  <ن֨Խͨ͠΋ͷ> ≃ 1 σ(b−1γ)2 exp ( − 1/r b−1γ) ( 1 r ) 3 exp ( − b rγ ) ∝ r for a small r : for a large r : (PSPLIPWTLJ4FWFMJFW  ͷղ<ස౓Ұఆ> J. Phys. D: Appl. Phys. 41 (2008) 085405 M 0 0.2 0.4 0.6 0.8 1 1.2 size, r 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 f(r) f(r) Modelling of the evolution equation α distribution - Gaussian (0.7; 0.07) initial radius distribution - Gaussian (1; 0.1) frequency = 100 Hz non-dimensional times: 0; 1.1; 3.3; 4.4; 6.6; 7.7 0 0.2 0.4 0.6 0.8 1 size,r Solution of the Fokker–Planck eq <ln(α)>=–0.36; <ln**2(α)>=0.14 initial radius distribution - gaussia frequency=100 Hz non-dimensional times: 0; 1.1; 3.3 (Gorokhovski & Seveliev 2008)
  32. ଎౓࿦ ܦ࣌มԽ ͱͷରԠ ( ln α νt) z2k+1 ( ln2

    α νt)2k+1 = t→∞ k(2k + 1)!! ln2 α νt ln3 α 3 ln2 α 1 + O 1 t , (12) where O(1/t) denotes smaller terms of order 1/t, and lnl α = 1 0 lnl αq(α) dα. In these expressions, we use ( )!! to denote the semi-factorial function (2k − 1)!! = 1 · 3 · 5 · . . . · 2k − 1, (2k + 1)!! = 1 · 3 · 5 · . . . · (2k + 1), it being understood that (−1)!! = 1! = 1. Comparing of moments (11), (12) with even and odd moments of the Gaussian distribution G(z) = (1/ √ 2πσ)e−z2/2σ2 , which are z2k /σ2k = (2k − 1)!! and z2k+1 /σ2k+1 = 0, shows that for large times, the central logarithmic moments of the solution of equation (6) tend to moments of the Gaussian distribution. As a result, the asymptotic solution of equation (6) can be derived: f (r, t) = t→∞ 1 R 1 2π ln2 α νt exp − ln α 2 2 ln2 α νt × exp − (ln(r/R))2 2 ln2 α νt R r 1−( ln α / ln2 α ), (13) where R is the initial length scale, R = e ln r t0 . This expression confirms the main result of Kolmogorov [1] 2 ln α r As time tends to infinity, this distribution tends to the pow distribution: f (r, t) ∼ t→∞ 1 r 1−( ln α / ln2 α ) . (1 It should be noted that the power distribution (17) is only va for r > rcr(t), where rcr(t) is a lower boundary of leng scales, which decreases exponentially with time: rcr(t) → when t → ∞. So that the power distribution of the form f (r, t|rcr) = βrβ crr−1−β, r > rcr, 0, r < rcr, with β > 0, tends to the Dirac delta function as time goes infinity: f (r, t|rcr) → rcr→0 δ(r). This result follows from the fact that βrβ cr ∞ rcr r−1−β dr = 1 Let us make another important remark. Previous stud (e.g. [10,11]) stated that when the dispersion (ln r − ln r ) is large, the ratio ln r / (ln r − ln r )2 is small an therefore, the log-normal distribution could be mimicked by 1/r distribution. However, one can show that first logarithm moments of the solution of equation (6) evolve as <f ͸F ͷن֨Խ> (Gorokhovski & Seveliev 2008) ≃ 1 σ(b−1γ)2 exp ( − 1/r b−1γ) ( 1 r ) 3 exp ( − b rγ ) ∝ r for a small r : for a large r : γ ⟶ νt b ⟶ α ૉࡐͷཁҼ ֎తཁҼ ύϥϝʔλͷରԠ (PSPLIPWTLJ4FWFMJFW  ͷղ<ස౓Ұఆ> S N (r) = b3 σγ3 1 r4 1 eb/rγ − 1 "PLJ'VSVF  <ن֨Խͨ͠΋ͷ> J. Phys. D: Appl. Phys. 41 (2008) 085405 M 0 0.2 0.4 0.6 0.8 1 1.2 size, r 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 f(r) f(r) Modelling of the evolution equation α distribution - Gaussian (0.7; 0.07) initial radius distribution - Gaussian (1; 0.1) frequency = 100 Hz non-dimensional times: 0; 1.1; 3.3; 4.4; 6.6; 7.7 0 0.2 0.4 0.6 0.8 1 size,r Solution of the Fokker–Planck eq <ln(α)>=–0.36; <ln**2(α)>=0.14 initial radius distribution - gaussia frequency=100 Hz non-dimensional times: 0; 1.1; 3.3 (Gorokhovski & Seveliev 2008) ؀ڥ৔ΤωϧΪʔͷ૿Ճ
  33. ·ͱΊ w ؀ڥ৔ͱϚΠΫϩϓϥενοΫͷഁյΛͭͳ͙෺ཧϞσϧΛ࣍ͷೋͭͷݪཧʹج͍ͮͯఏҊ͠ ͨɻ ᶃ খ͞ͳഁยͷܗ੒΄Ͳେ͖ͳഁյΤωϧΪʔΛཁ͢Δ ᶄ ഁյΤωϧΪʔͷੜى֬཰͸ϘϧπϚϯ෼෍ʹै͏ w ຊϞσϧ͔Βಋ͔ΕΔཻܘ෼෍͸ϓϥϯΫ෼෍ͱྨࣅ͠ɺ͞Βʹɺ؍ଌ͞Εཻͨܘ෼෍ΛΑ͘

    આ໌͢Δɻ w ྼԽ౓͕ෆۉҰͰ͋ͬͨΓɺαΠζґଘ͢Δ৔߹ʹ΋ɺຊϞσϧ͸ద༻Ͱ͖ΔՄೳੑ͕͋Δɻ w ຊϞσϧ͸ฏߧ౷ܭྗֶʹج͍͍ͮͯΔ͕ɺഁյͷܦ࣌มԽ΋಺แ͍ͯ͠ΔՄೳੑ͕͋Δɻ Aoki K., and R. Furue (2021): A model for the size distribution of marine microplastics: a statistical mechanics approach, PLOS ONE ࠓ೔ͷ࿩͸ͪ͜ΒͰ΋͝ཡʹͳΕ·͢ˣ